math problem solving examples grade 6

• 6th Grade Math
• Solving word problems

Grade 6 Math Word Problems: Tips, Tricks, and Answers

Do you want to stimulate your 6 th grader’s creative thinking skills? Then, enjoy this brilliant math article. In this comprehensive guide, we will provide you with a treasure trove of fun solving strategies, tips, tricks, and answers to tackle those tricky grade 6 math word problems that have been confusing your students for some time now.

In this page, you will discover why math word problems are important for 6th-graders and simple methods of breaking down complex word problems into manageable steps.

Nevertheless, we will introduce you to Mathskills4kids.com , an outstanding website with thousands of common types of grade 6 math word problems and a step-by-step approach to solving them. Interestingly, we will illustrate how to use diagrams and models to solve math word problems efficiently.

Learn to love Grade 6 Math word problems with these worksheets and answers

Hello and welcome to Grade 6 Math word problems worksheets and answers , where your 6 th Grade students will learn to love and solve math problems and activities at all times.

We understand that word problems can often frustrate students, as they require a solid understanding of mathematical concepts and the ability to interpret and apply them to real-life situations. That's why we have compiled a collection of proven strategies and techniques to empower your students to approach word problems confidently and accurately.

From understanding problem-solving strategies to breaking down complex questions into manageable steps, this guide will equip you with the knowledge and resources to make math word problems a breeze. So, let's dive in and unlock the secrets to conquering grade 6 math word problems together!

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Why are math word problems important for 6th-graders.

Math word problems are about more than just doing calculations. They are also about applying your math knowledge to real-world situations. Math word problems are important for 6 th graders because they help them to:

• Learn how to use different math concepts and skills in various contexts
• Develop their logical thinking and reasoning abilities
• Enhance their communication and literacy skills
• Prepare them for standardized tests and future math courses

Math word problems also make math more exciting and relevant. They show us how math can solve everyday problems and challenges. They also expose us to different topics and scenarios we may not encounter in our regular math lessons.

Strategies for solving Grade 6 math word problems

Solving Grade 6 math word problems can be intimidating, especially involving multiple steps or operations. But don't worry. Some general strategies will help your students confidently approach any word problem. Here are some of them:

• Please encourage them to read the problem carefully and identify the given information, the unknowns, and the question.
• They should rewrite the problem in their own words or summarize it in a sentence.
• Let them choose a suitable method or strategy to solve the problem. Some standard methods are guessing and checking, making a table or chart, drawing a picture or diagram, using a formula or equation, working backward, or using logical reasoning.
• They must show their work and explain each step clearly. Use appropriate units, labels, symbols, and terms.
• Lastly, tell them to check their answer by plugging it back into the problem or using a different method. Ensure their answer makes sense and answers the question.

Breaking down complex word problems into manageable steps

Some word problems may seem too complex or confusing at first glance. They may have too much information, too many steps, or too many operations. In such cases, breaking down the problem into smaller and simpler parts is helpful. Here are some tips on how to do that:

Words like "difference," "subtract," "take away," or "minus" indicate s ubtraction .

Words like "product," "multiply," "times," or "of" indicate multiplication .

Words like "quotient," "divide," "per," or "out of" indicate division .

Words like "ratio," "fraction," "percent," or "part" indicate fractions or decimals .

Words like "equal," "same as," or "is" indicate equations .

Words like "more than,” “less than," "greater than," or "smaller than" indicate inequalities .

Words like "average," "mean," or "median" indicate statistics .

Words like "area," "perimeter," "volume," or "surface area" indicate geometry , etc.

• Use parentheses, brackets, or other symbols to group the parts of the problem that belong together . For example, if the problem says:

You can rewrite it as:

• (John has 12 apples) + (Mary has 8 apples) = (total number of apples) / (4 people) = (number of apples per person)

This way, you can see the structure of the problem more clearly and focus on one part at a time.

• Solve each part of the problem separately and write down the intermediate results . For example, using the previous problem:
• (John has 12 apples) + (Mary has 8 apples) = (total number of apples)
• 12 + 8 = 20
• (total number of apples) / (4 people) = (number of apples per person)

This way, you can keep track of your work and avoid making mistakes.

• Combine the intermediate results to get the final answer. For example, using the previous problem :

This way, you can answer the question and check your answer.

Common types of grade 6 math word problems

There are many types of word problems that you may encounter in grade 6 math . Some of the most common ones found on Mathskills4kids.com are:

• Ratio and proportion problems : These problems involve finding the relationship between two quantities with the same unit or measure. For example, if 12 pencils cost $3, how much do 20 pencils cost?
• Percent problems : These problems involve finding the part, whole, or percent of a quantity. For example, if 30% of a class is boys, and there are 24 students, how many boys are there?
• Fraction problems : These problems involve adding, subtracting, multiplying, or dividing fractions. For example, how much pizza do you have left if you have 2/3 of a pizza and eat 1/4?
• Decimal problems : These problems involve adding, subtracting, multiplying, or dividing decimals. For example, if you buy a shirt for $12.50 and a pair of jeans for $24.75, how much change do you get from $50?
• Measurement problems : These problems involve converting between different units of measurement or finding the perimeter, area, or volume of shapes. For example, if a rectangle has a length of 15 cm and a width of 10 cm, what is its area in square meters?
• Algebra problems : These problems involve finding the value of an unknown variable or expression. For example, if x + 5 = 13, what is the value of x?

A step-by-step approach to solving different types of word problems

No matter what type of word problem your 6 th graders are faced with, they can use the following steps to solve it:

• They should read the problem carefully and identify the given information and the question.
• They must choose a variable to represent the unknown quantity and write an equation or expression that relates the given information and the question.
• They can now solve the equation or expression and find the variable's value.
• They can check their answer by plugging it back into the equation or expression to see if it makes sense.
• They can now write their answer in complete sentences and include the appropriate units.

Using diagrams and models to solve math word problems

Sometimes, it can be helpful to use diagrams and models to visualize the problem and make it easier to solve. Some examples of diagrams and models for solving math word problems are:

• Tape diagrams : These are horizontal or vertical bars showing the relationship between two quantities. For example, you can use a tape diagram to show how much money each person gets when $60 is shared equally among 4 people.
• Number lines : These horizontal lines show numbers and their relative positions. For example, you can use a number line to show how to add or subtract fractions with different denominators.
• Tables : These are grids that show data in rows and columns. For example, you can use a table to show how to find equivalent ratios or fractions.
• Charts : These are graphical representations of data using bars, circles, lines, or other shapes. For example, you can use a chart to show how to find the percent of a quantity or compare different quantities.
• Drawings : These are sketches or illustrations that show shapes or objects. For example, you can use a drawing to show how to find a shape's perimeter, area, or volume.

Providing answers and explanations to sample Mathskills4kids’ Grade 6 math word problems

Here are some sample Grade 6 math word problems with answers and explanations available at Mathskills4kids.com :

Answer : 6 cups of flour

Explanation : This is a ratio problem. We can use a tape diagram to show the relationship between flour and sugar.

Flour →|<---2 cups--->|<---2 cups--->|<---2 cups---> = 6

Sugar→|<---3 cups--->|<---3 cups--->|<---3 cups---> = 9

We can see that for every 3 cups of sugar, we need 2 cups of flour. So, for 9 cups of sugar, we need 6 cups of flour.

Answer : $15

Explanation : This is a percent problem. We can use a formula to find the sale price of the shirt.

Sale price = Original price - Discount

Discount = Percent off x Original price

We know the percent off is 40%, and the original price is $25. So, we can plug these values into the formula and solve for the sale price.

Discount = 40% x $25

Discount = 0.4 x $25

Discount = $10

Sale price = $25 - $10

Sale price = $15

Tips for improving problem-solving skills in Grade 6 math

Here are some tips that can help students improve their problem-solving skills in grade 6 math :

• Please encourage them to practice regularly and try different types of word problems.
• They should review the concepts and skills they have learned and apply them to new situations.
• They can use different strategies and methods to solve word problems and compare their results.
• Let them ask for help from the teacher, parents, or peers if they get stuck or confused.
• They should learn from their mistakes and try to avoid them in the future.

Bonus: additional resources to reinforce Grade 6 math problem skills

If you want to enhance your student's Grade 6 math word problem skills , or if you need some extra help, here are some additional and useful web links that you can check out:

• Math Playground : This website has a lot of fun and interactive games that let 6 th graders practice different types of math word problems, such as fractions, decimals, ratios, proportions, and more. They can also watch videos explaining how to solve some problems. https://www.mathplayground.com/wordproblems.html .
• Khan Academy : This website has many videos and exercises covering various topics in Grade 6 math, including word problems. Students can learn at their own pace and track their progress. https://www.khanacademy.org/math/cc-sixth-grade-math .
• IXL : This website has many practice questions aligned with the Common Core standards for Grade 6 math. Students can choose from different categories of word problems, such as expressions and equations, geometry, statistics, and more. They can also get instant feedback and explanations for their answers. https://www.ixl.com/math/grade-6 .
• Math Goodies : This website has a lot of worksheets and lessons that teach 6 th graders how to solve different types of word problems, such as percent, ratio, proportion, and more. They can also find tips and tricks for solving word problems faster and easier. https://www.mathgoodies.com/math-mammoth/worksheets/pdf/grade_6_word_problems.pdf .

Thank you for sharing the links of MathSkills4Kids.com with your loved ones. Your choice is greatly appreciated.

Math word problems can be challenging for many 6th-graders, but they are also important for developing their mathematical thinking and reasoning skills.

Using the strategies and steps we discussed in this article, your student's ability to solve different types of word problems in Grade 6 math can be improved.

You can also use the diagrams and models we have shown you to help your students visualize the problem and find the solution. Encourage them to practice as much as possible, and they must not be afraid to ask for help if needed.

You can also use the web links that we have provided to reinforce your student’s learning and have fun with math. We hope this article has helped your 6th-grade students feel more confident and prepared for tackling Grade 6 math word problems.

Happy problem-solving!

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Free Printable Math Word Problems Worksheets for 6th Grade

Math Word Problems: Discover a vast collection of free printable worksheets for Grade 6 students, created by educators to enhance their mathematical skills and problem-solving abilities. Dive into the world of numbers with Quizizz!

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Explore printable Math Word Problems worksheets for 6th Grade

Math Word Problems worksheets for Grade 6 are an essential resource for teachers who want to help their students develop strong problem-solving skills and a deep understanding of mathematical concepts. These worksheets provide a variety of engaging and challenging problems that require students to apply their knowledge of math in real-world situations. With a wide range of topics covered, including fractions, decimals, percentages, and geometry, Grade 6 Math Word Problems worksheets are designed to align with the Common Core State Standards and support teachers in their efforts to create a well-rounded math curriculum. By incorporating these worksheets into their lesson plans, teachers can ensure that their students are receiving the practice and reinforcement they need to excel in math.

Quizizz is an excellent platform for teachers to access a vast library of Math Word Problems worksheets for Grade 6, along with other valuable resources to enhance their students' learning experience. This interactive platform offers a variety of features, such as customizable quizzes, real-time feedback, and gamification elements, to keep students engaged and motivated. In addition to Grade 6 Math worksheets, Quizizz also provides resources for other subjects and grade levels, making it a one-stop-shop for teachers looking to diversify their instructional materials. By utilizing Quizizz in conjunction with Math Word Problems worksheets for Grade 6, teachers can create a dynamic and effective learning environment that fosters a love for math and sets their students up for success.

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6th Grade Math Worksheets Hub Page

Welcome to our 6th Grade Math Worksheets hub page.

This is a new hub page and currently under development - so there will be more 6th grade resources on the way soon!

Here you will find a wide range of free printable worksheets that follow the standards for 6th Grade .

Come and take a look at our adding subtracting fractions page, or our 6th grade math games page. Looking for some help with algebra? We have some basic algebra worksheets too.

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• This page contains links to other Math webpages where you will find a range of activities and resources.
• If you can't find what you are looking for, try searching the site using the Google search box at the top of each page.

6th Grade Math Learning

Here are the standards for the end of the year for 6th Grade.

• Find the greatest common factor of whole numbers up to 100.
• Find the least common multiple of two whole numbers up to 12.
• Recognise opposite signs of numbers (positive and negative) as being on opposite sides of 0 on a number line.
• Recognise that the opposite of the opposite of a number is the number itself e.g. -(-7) = 7
• Understand a rational number as a point on a number line.
• Find and position rational numbers on a range of number lines.
• Understand the absolute value of a number.
• Write, interpret and explain statements about rational numbers, such as -3°C > -7°C
• Write and evaluate expressions involving whole number exponents.
• Fluently divide multi-digit numbers.
• Fluently add, subtract, multiply and divide multi-digit decimals using the standard algorithm.
• Find the percentage of a quantity as a rate per 100.
• Divide fractions by fractions using visual models.
• Solve problems involving dividing fractions by fractions.
• Understand the concept of ratio and use ratio language.
• Understand the concept of a unit rate a/b associated with a ratio a:b.
• Use ratio and rate reasoning to solve a range of problems.
• Write, read, and evaluate expressions in which letters stand for numbers
• Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient);
• Evaluate expressions at specific values of the variables of different formulas.
• Apply the properties of operations to generate equivalent expressions, e.g. 10 + 5y = 5(2 + y).
• Reason about and solve one-variable equations and inequalities.
• Write an inequality of the for x<c or x>c to represent a constraint or condition.
• Represent and analyze quantitative relationships between dependent and independent variables.
• Develop understanding of statistical variability
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• Display numerical data in plots on a number line, including dot plots, histograms, and box plots
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• Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.
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• Use the nets to find the surface area of these figures

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6th Grade Place Value Zone

Ordering numbers, including negative numbers, decimals to 3dp and rational numbers.

The sheets in this section involve negative numbers, decimals to 3 decimal places, and a range of rational numbers.

There are sheets with decimals up to 10, and also sheets with numbers from -10 to 10.

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Here you will find a range of 6th Grade Number Worksheets covering skills connected to Numbers and the Number System.

Using these sheets will help your child to:

• understand how to use inequalities including where you have two values
• understand how to use exponents (powers) of a number;
• find the greatest common factors of two numbers up to 100
• find the least common multiple of two numbers up to 12.
• extend their knowledge of prime and composite (non-prime) numbers up to 100;
• know and be able to use the PEMDAS (or PEDMAS) rule.
• understand and use absolute value

Inequalities

• Inequalities on a Number Line
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Here you will find a range of printable mental math 6th grade quizzes for your child to enjoy.

Each worksheet tests the children on a range of math topics from number facts and mental arithmetic to geometry, fraction and measures questions.

A great way to revise topics, or use as a weekly math quiz!

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Decimal counting worksheets.

Using these sheets will support you child to:

• count on and back by multiples of 0.1;
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• count on and back into negative numbers.
• find number bonds to 1;
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6th Grade Addition & Subtraction Worksheets

The following 6th grade math worksheets involve using addition and subtraction with a range of different number types.

• add and subtract decimals up to 3dp;
• add a columns of multi-digit numbers, including decimals.
• add and subtract positive and negative integers

The addition worksheet generator below will create a range of addition problems in columns, including decimals.

• Addition With Regrouping Worksheet Generator
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• Adding Positive and Negative Numbers (randomly generated)
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6th Grade Multiplication Worksheets

Here you will find a range of Free Printable 5th Grade Multiplication Worksheets.

The following worksheets involve using the 6th Grade Math skills of multiplying multi-digit numbers, including decimals.

• extend their knowlege of multiplication to decimals;
• use their multiplication tables to answer related facts, including decimals;
• multiply a range of decimals by a whole number;
• multiply positive and negative numbers.
• Multiplying Decimals by Whole Numbers
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6th Grade Division Worksheets

Here you will find a range of Free Printable 6th Grade Division Worksheets.

The following worksheets involve using the 6th Grade Math skills of dividing multi-digit numbers, including decimals, and solving division problems.

Using these sheets will help your child learn to:

• divide multi-digit numbers by one and 2-digit numbers;
• divide decimals.
• Divisibility Rules Worksheets
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6th Grade Math Problems

Here you will find our selection of free 6th grade math word problems.

Each sheet is availabel in both standard and metric units (where applicable).

Each sheet comes complete with a separate answer sheet.

All the problems are based around 'real life' such as the planets, heights of mountains, or length of rivers.

Using these sheet will help your child to:

• apply their addition, subtraction, multiplication and division skills;
• apply their knowledge of rounding and place value;
• solve a range of problems including "real life" problems and ratio problems.
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6th Grade Fraction Worksheets

Here you will find a range of free printable 6th Grade Fraction Worksheets.

At 6th Grade level, children are introduced to adding and subtracting fractions with different denominators. They know and can use equivalent fractions, and can multiply and divide fractions by whole numbers, as well as mixed numbers.

• add and subtract fractions and mixed numbers;
• understand how to multiply fractions by a whole number;
• understand how to multiply two fractions together, including mixed fractions;
• understand the relationship between fractions and division;
• know how to divide fractions and mixed fractions;
• convert decimals to fractions.
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• Multiplying Mixed Fractions
• How to Divide Fractions
• Dividing Fractions by Whole numbers
• Divide Whole numbers by Fractions
• How to Divide Mixed Numbers
• Multiplying and Dividing Fractions (Randomly Generated)
• Add Subtract Multiply Divide Fractions (Randomly Generated)
• Free Printable Fraction Riddles (harder)
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6th Grade Percentage Worksheets

Take a look at our percentage worksheets for finding the percentage of a number or money amount.

We have a range of percentage sheets from quite a basic level to much harder.

• Percentage of Numbers Worksheets
• Money Percentage Worksheets
• Basic Percentage Word Problems (easier)
• 6th Grade Percent Word Problems (harder)

6th Grade Ratio and Unit Rate Worksheets

These 5th grade ratio worksheets are a great way to introduce this concept.

We have a range of part to part ratio worksheets and slightly harder problem solving worksheets.

• Ratio Part to Part Worksheets
• Ratio and Proportion Worksheets
• The Definition of Unit Rate
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6th Grade Algebra Worksheets

If you are looking for some 6th grade algebra worksheets to use with your child to help them understand simple equations then try our selection of basic algebra worksheets.

There are a range of 6th grade math worksheets covering the following concepts:

• Generate the algebra - and write your own algebraic expressions;
• Calculate the algebra - work out the value of different expressions;
• Solve the algebra - find the value of the term in the equation.
• Use the distributive property to factorize and expand different expressions
• 6th Grade Distributive Property Worksheets
• Expressions and Equations 6th Grade
• Basic Algebra Worksheets (6th & 7th Grade)
• Input and Output Function Tables with Algebraic Functions

Geometry & Measurement Zone

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6th Grade Geometry Worksheets

Here is our range of 6th grade Geometry worksheets.

• Geometry Nets Information and Worksheets
• Parts of a Circle Worksheets
• Coordinate Plane Worksheets (All 4 Quadrants)

6th Grade Area & Perimeter Worksheets

• Area of Parallelogram Worksheets
• Area of Quadrilaterals
• Area of Right Triangle
• Square Inside a Circle Area Support Page
• Surface Area Worksheet 6th Grade

6th Grade Measurement Worksheets

• Metric Conversion Worksheets (4th to 6th Grade)
• Convert Customary Units of Length Worksheets (randomly generated)
• Convert Customary Units of Capacity Worksheets (randomly generated)

Data & Statistics Worksheets

Find links to our 6th grade Statistics worksheets below.

Using these 6th grade math worksheets will help you to:

• find the mean of up to 5 numbers;
• find a missing data point when the mean is given.
• find the median of a set of data.
• create and interpret line plots.
• create and interpret dot plots and box plots.
• Median Worksheets
• Mean Worksheets
• Mode and Range Worksheets
• Mean Median Mode and Range Worksheets
• Line Graphs 6th Grade Worksheets
• Dot Plot Worksheets
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Fun Zone: Puzzles, Games and Riddles

• 6th Grade Math Games

Here you will find a range of free printable 6th Grade Math games.

All children like to play Math games, and you will find a good range of Grade 6 Math Games here for your child to play and enjoy.

The following games involve different 5th Grade Math activities which you and your child can enjoy together.

• Algebra Math Games
• 6th Grade Math Puzzles

Here you will find a range of printable 6th grade math puzzles for your child to enjoy.

The puzzles will help your child practice and apply their addition, subtraction, multiplication and division facts as well as developing their thinking and reasoning skills in a fun and engaging way.

Using these puzzles will help your child to:

• learn and practice their addition and subtraction facts, including fractions and decimals;
• develop their understanding of negative numbers;
• practice and apply multiplication and division facts;
• develop problem solving skills and reasoning.

6th Grade Math Quiz

Here is our collection of Math Quizzes for 6th grade.

Quizzes are a great way to practise math skills or to asses knowledge.

• 6th Grade Math Quiz page

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" class="arrow-title-img"> Grade 6 math word problems with solutions and explanations

• INTRODUCTION
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6th grade math word problems curriculum.

Grade 6 math word problems with solutions and explanations is a complete training program made of games, worksheets, lessons for 6th graders on addition and subtraction word problems, fractions and mixed numbers word problems, number theory word problems, multiplication and division word problems, mixed operation word problems, solving and estimation word problems, etc.

So, we invite you to quickly grab the best of 6th Grade math word problems which consist of all the concepts in grade 6 math. Of equal interest to generate your kid’s eagerness and love for solving these math problems, we have designed very captivating math word problem resources for grade 6 .

In this light, your young math learners will significantly enhance their understanding of how to confidently apply math skills to all 6th Grade math word problems curricula. In addition, we have devised plenty of fun strategies useful for quick and effective solving all real-life math problems.

• Addition And Subtraction Online Practice And Worksheets
• Add and subtract whole numbers word problems
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• Divide fractions 6th grade
• Divide fractions and mixed numbers 6th grade
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• Divisibility rule skills for Grade 6
• Division of numbers word problems
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• Adding and subtracting fractions with like denominators word problems
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• Add subtract multiply or divide two fractions
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• Write numbers from digits to words 6th grade online practice
• Write numbers from words to digits in grade 6

How 6th Grade math word problems enhance kids' math skills?

Most often than not, we begin to wonder how 6th-grade math word problems enhance kids’ math skills ! But after engaging in full practice of solving these grade 6 math word problems, you’ll be surprised at your kid’s new enriching math skills.

First and foremost, as we have selected problems from all areas of our daily lives relating to addition, subtraction, multiplication, division, fractions, measurements, etc., your kids will gain more expansive reasoning to determine how best to approach any real-life math problem quickly.

Moreover, as we have designed simple solving steps and tips for solving these problems, your kids will become accustomed to the solving process, thus gaining math fluency and skills.

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Free Math Worksheets — Over 100k free practice problems on Khan Academy

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Grade 6 Algebra Word Problems

These lessons look at solving Grade 6 algebra word problems by writing equations and expressions that matches the situation? The following are some examples and solutions for algebra word problems that you will commonly encounter in Grade 6.

Related Pages More Math Word Problems Algebra Word Problems Algebra Word Problems

Grade 6 Algebra Word Problems How to write one-step equations for grade 6 algebra word problems?

Example 1: Anna wants to celebrate her birthday by eating pizza with her friends. For $42.50 total, they can buy p boxes of pizza. Each box of pizza costs $8.50. Write an equation that matches the situation.

Example 2: Mr. Herman’s class is selling candy for a school fundraiser. The class has a goal of raising $500 by selling c boxes of candy. Foe every box they sell, they make $2.75. Write an equation that the students could solve to figure out how many boxes of candy they need to sell.

Grade 6 Algebra Word Problems How to write algebraic expressions from word problems?

Example 1: The price of a visit to the dentist is $50. If the dentist fills any cavities, an additional charge of $100 per cavity gets added to the bill If the dentist finds v cavities, what will the cost of the visit be? Write your answer as an expression.

Example 2: Sunny earns $12 per hour delivering cakes. She worked for x hours this week. Unfortunately, she was charged $15 for a late delivery on Tuesday. How much money did Sunny earn this week? Write your answer as an expression.

Example 3: There are c players on the Cougars hockey team. The team scored a total of 36 goals this season. One of the players, Matthew, scored 2 more goals than the average per player. How many goals did Matthew score? Write your answer as an expression.

Example 4: Hannah has 127 books in her collection. Her school is hosting a book donation. There are z students at her school, and they each plan to donate the same amount of books and reach a total donation of 300 books. How many books will Hannah have in her collection after her donation. Write your answer as an expression.

Grade 6 Algebra Word Problems - Rate, Distance, Time

Example 1: Suzie ran a race. She ran 5 miles per hour, and the race took t hours to complete. How long was the race? Write your answer as an expression.

Example 2: The Running Aces card team won $548 playing in poker tournaments last year. The winnings were split among the p players. How much money did each player receive? Write your answer as an expression.

Example 3: Phil received a prize of x dollars from a poker tournament. The tournament cost him 100 dollars to enter. What were Phil’s net winnings from the tournament? Write your answer as an expression.

Example 4: Hillary made 48 chocolate chip cookies and y sugar cookies How many total cookies did Hillary make? Write your answer as an expression.

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Free Mathematics Tutorials

Equations in one variable- grade 6.

Grade 6 examples and questions on equations and problems in one variable with detailed solutions , at the bottom of the page, and explanations are presented. If you find that some of the questions are challenging do not skip them, spend time on them and work in groups. We learn math by solving challenging questions.

Questions and Problems

• (x - 8) / 3 = 9
• (2x +7) / 3
• x/2 + 4 = 6
• x = 2 and 2 x = 4
• x + 3 = 6 and x + 4 = 8
• x / 2 = 2 and x = - 4
• 3 x = 9 and x + 1 = 4
• Jimmy, Toby and Dina contributed a total of $123 to buy a gift for their mother. Of the $123, Jimmy contributed $34 and Dina $45. How much money did Toby contribute for the gift?

Solutions to the Above Questions and Problems

• x = 0 left side: 2 x - 4 = 2(0) - 4 = 0 - 4 = - 4 right side: 4 The two sides are not equal and therefore x = 0 does not satisfy the given equation.
• x = 4 left side: 2 x - 4 = 2(4) - 4 = 8 - 4 = 4 right side: 4 The two sides are equal and therefore x = 4 satisfies the given equation and it is called a solution to the equation. Only one value of x may satisfy the given equation and therefore there is no need to check the remaining values of x , they will not satisfy the given equation.
• x = - 3 left side: x / 3 - 1 = 2 = (-3) / 3 - 1 = - 1 - 1 = - 2 right side: 2 The two sides are not equal and therefore x = - 3 does not satisfy the given equation.
• x = 6 left side: x / 3 - 1 = 6 / 3 - 1 = 2 - 1 = 1 right side: 2 The two sides are not equal and therefore x = 6 does not satisfy the given equation.
• x = - 9 left side: x / 3 - 1 = (- 9) / 3 - 1 = - 3 - 1 = - 4 right side: 2 The two sides are not equal and so x = - 9 does not satisfy the given equation.
• x = 9 left side: x / 3 - 1 = (9) / 3 - 1 = 3 - 1 = 2 right side: 2 The two sides are equal and so x = 9 satisfies the given equation and is a solution.
• Solve x - 6 = 12 add +6 to both sides x - 6 + 6 = 12 + 6 Simplify x = 18 , solution to the given equation.
• Solve 3 = x + 3 subtract 3 from both sides of the equation 3 - 3 = x + 3 - 3 Simplify 0 = x , solution to the given equation.
• Solve 2 + x = 8 subtract 2 from both sides of the equation 2 + x - 2 = 8 - 2 Simplify x = 6 , solution to the given equation.
• Solve 2 x = 16 Divide both sides of the equation by 2 2 x / 2 = 16 / 2 Simplify x = 8 , solution to the given equation.
• Solve x / 3 = 5 Multiply both sides of the equation by 3 3 (x / 3) = 3 (5) Simplify x = 15 , solution to the given equation.
• Solve the two equations x = 2 and 2 x = 4 First equation: x = 2 is solved Second Equation: Divide both sides of the equation by 2 and simplify 2 x / 2 = 4 / 2 gives x = 2 The two equations have the same solutions
• Solve the two equations x + 3 = 6 and x + 4 = 8 First equation: x + 3 = 6 ; subtract 3 from both sides and simplify x + 3 - 3 = 6 - 3 gives x = 3 Second Equation: x + 4 = 8 ; subtract 4 from both sides and simplify x + 4 - 4 = 8 - 4 gives x = 4 The two equations do not have the same solutions.
• Solve the two equations x / 2 = 2 and x = - 4 First equation: x / 2 = 2 ; multiply by both sides by 2 and simplify 2(x / 2) = 2(2) gives x = 4 Second Equation: x = - 4 is already solved The two equations do not have the same solutions.
• Solve the two equations 3 x = 9 and x + 1 = 4 First equation: 3 x = 9 ; divide by both sides by 3 and simplify 3 x / 3 = 9 / 3 gives x = 3 Second Equation: x + 1 = 4 ; subtract 1 from both sides and simplify x + 1 - 1 = 4 - 1 gives x = 3 The two equations have the same solutions.
• Solution The value of x that makes 2 x + 6 equal to 12 is the solution to the equation 2 x + 6 = 12 Subtract 6 to both sides and simplify 2 x + 6 - 6 = 12 - 6 2x = 6 Divide both sides of the equation by 2 and simplify 2 x / 2 = 6 / 2 gives x = 3 Check by substituting x by 3 in the given expression 2 x + 6 = 2 (3) + 6 = 6 + 6 = 12 which is equal to 12. x = 3 makes 2 x + 6 equal to 12.
• Solution The value of x that makes 4 x + 6 equal to 2 + 12 is the solution to the equation 4 x + 6 = 2 + 12 Simplify the right side 4 x + 6 = 14 Subtract 6 to both sides and simplify 4 x + 6 - 6 = 14 - 6 4x = 8 Divide both sides by 4 and simplify 4 x / 4 = 8 / 4 gives x = 2 Check by substituting x by 2 in the expression 4 x + 6 4 x + 6 = 4 (2) + 6 = 8 + 6 = 14 which is equal to right side 2 + 12. x = 2 makes 4 x + 6 equal to 2 + 12 .
• Solution The sum is represented by + operation in math. Hence the phrase "The sum of d and 23" is represented by d + 23 and "is 56" means is equal to 56. Hence the statement "the sum of d and 23 is 56" is represented by the equation d + 23 = 56 and to find d, we need to solve the equation above. Subtract 23 from both sides of the equation d + 23 - 23 = 56 - 23 Simplify and solve for d d = 33 Check the answer to the question. d + 23 = 33 + 23 = 56 "The sum of d (= 33) and 23 is 56" is correct.
• Solution The phrase "Seven subtracted from x" is represented by x - 7 and "is 41" means is equal to 41. Hence the statement "Seven subtracted from x is 41" is represented mathematically by the equation x - 7 = 41 We find x by solving the equation above. Add 7 to both sides of the equation x - 7 + 7 = 41 + 7 Simplify and solve for x x = 48 Check the answer to the question. 48 - 7 = 41 "Seven subtracted from x( = 48) is 41" is correct.
• Solution The phrase "The product of y and 6" is represented by 6 × y = 6 y and "is 36" means is equal to 36. Hence the statement "The product of y and 6 is 36" is represented mathematically by the equation 6 y = 36 y is found by solving the equation above. Divide both sides of the equation by 6. 6 y / 6 = 36 / 6 Simplify and solve for y y = 6 Check the answer to the question. 6 × 6 = 36 "The product of y ( = 6) and 6 is 36" is correct.
• Solution The phrase "division of b by 5" is represented by b ÷ 5 and "is 4" means is equal to 4. Hence the statement "The division of b by 5 is 4" is represented mathematically by the equation b ÷ 5 = 4 Multiply both sides of the equation by 5. 5 (b ÷ 5) = 5 × 4 Simplify and solve for b b = 20 Check the answer to the question. b ÷ 5 = 20 ÷ 5 = 4 "The division of b( = 20) by 5 is 4" is correct.
• Solution Jacky has x cards and Jimmy has 23 cards. Jacky: x cards Jimmy: 23 cards and "together they have 121 cards" means the total (sum) number of cards of both. Hence the two statements "Jacky has x cards and Jimmy has 23 cards" and "together they have 121 cards" is represented mathematically by the equation x + 23 = 121 Subtract 23 from both sides of the equation above. x + 23 - 23 = 121 - 23 Simplify and solve for x x = 98 Jacky has x cards which was found to be equal to 98 Check the answer to the problem by Jacky's and Jimmy's cards 98 + 23 = 121 The answer x = 98 is correct because when the cards are added they give a total of 121.
• Solution The gift bought cost $123 which is the total contributions of all three. Hence Total : 123 We know what Jimmy and Dina contributed. Jimmy : $34 Dina : $45 We do not know what Toby contributed and therefore this is the unknown in this problem. Hence let us give it a name using the letter c, for example, to the amount contributed by Toby. Toby : c They put all their money together to buy the gift. Hence the contributions of all three is represented by the sum 34 + 45 + c All the money put togther was used to buy the gift which we know its cost $123; hence the equation 34 + 45 + c = 123 Simplify the left side and rewrite the equation as c + 79 = 123 Subtract 79 from both sides of the equation. c + 79 - 79 = 123 - 79 Simplify and solve. c = 44 which is the contribution, in dollars, of Topy for the gift to his mother. Check the answer to the problem by adding all contributions. 44 + 34 + 45 = 123 The answer c = 44 is correct because when all contributions are added they give a total of 123 which is the price paid for the gift.

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Nonlinear system of equations

Here you will learn about a nonlinear system of equations, including what they are and how to solve them.

Students will first learn about a nonlinear system of equations as part of algebra in high school.

What is a nonlinear system of equations?

A nonlinear system of equations is two or more equations that form a system, where at least one equation is nonlinear.

One example of a nonlinear equation is a quadratic equation , which has variables that are raised to powers of 2, for example, x^2 and y^2.

There are other types of nonlinear equations, but this page will cover simple systems that involve a linear and a quadratic equation.

One key difference of a nonlinear system of equations compared to linear system of equations is that you can expect multiple answers . This is because of the way the graphs of linear and quadratic functions can intersect.

On the graph below, the straight line of the linear equation crosses the curved parabola of the quadratic equation at two points of intersection .

This means the system of equations has two valid answers .

This is an example of a nonlinear system of equations:

y=x^{2}-4 (quadratic)

y=x+2 (linear)

These points of intersection are the solutions to the system of equations. They are ordered pairs that satisfy all equations in the system.

Here, the solutions are x=3, \, y=5 and x=- \, 2, \, y=0. This can also be written as a coordinate pair (3, \, 5) and (- \, 2, \, 0).

What is a Nonlinear System of Equations?

Common Core State Standards

How does this relate to high school math?

• Algebra – Reasoning with Equations and Inequalities (HS.A.REI.C.7) Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y=- \, 3x and the circle x^{2}+y^{2}=3.

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How to solve nonlinear system of equations

To solve a set of system of equations you need to:

Use substitution to eliminate one of the variables.

Solve for the value(s) of one variable.

Use substitution to find the value(s) of the remaining variable.

Clearly state the final answer(s).

Check your answer by substituting both values into either of the original equations.

Nonlinear system of equations examples

Example 1: equation of a straight line and a quadratic (factor).

Use the fact that y=x+3 to substitute the value of y into the second equation.

Therefore x+3=x^{2}+5x-2.

Another way to think about this is that as both equations are equal to y, they must therefore be equal to one another.

2 Solve for the value(s) of one variable.

Since the equation is now quadratic, rearrange so that the equation is equal to 0.

Then solve:

x^{2}+4x-5=0 can be factored:

x+5=0 so x=- \, 5

x-1=0 so x=1

The two values of x are x=- \, 5 and x=1.

3 Use substitution to find the value(s) of the remaining variable.

The two values of x can be substituted into one of the original equations to find the two possible values of y.

Remember you can use either equation, so why not pick the easiest!

As y=x+3, when x=- \, 5, \, y=- \, 5+3=- \, 2.

As y=x+3, when x=1, \, y=1+3=4.

4 Clearly state the final answer(s).

x=- \, 5, \, y=- \, 2 and x=1, \, y=4.

5 Check your answer by substituting both values into either of the original equations.

Let’s use the quadratic equation to check the answers.

As y=x^{2}+5x-2, when x=- \, 5,

As y=x^{2}+5x-2, when x=1,

Both are valid so the answers are correct.

Let’s look at the graph of the system.

When graphed, these two equations intersect at (– \, 5, \, – \, 2) and (1, \, 4). So the solutions to the system of equations are:

Example 2: equation of a straight line and a quadratic (factor)

Use the fact that y=3x-12 to substitute the value of y into the second equation.

\begin{aligned}y&=3x-12 \\\\ y&=x^{2}-2x-6 \end{aligned}

Therefore 3x-12=x^{2}-2x-6.

x^{2}-5x+6=0 can be factored:

(x-3)(x-2)=0

x-3=0 so x=3

x-2=0 so x=2

The two values of x are x=3 and x=2.

As y=3x-12, when x=3, \, y=3(3)-12=- \, 3.

As y=3x-12, when x=2, \, y=3(2)-12=- \, 6.

x=2, \, y=- \, 6 and x=3, \, y=- \, 3.

As y=x^{2}-2x-6, when x=3,

\begin{aligned}y&=(3)^{2}-2(3)-6 \\\\ &=9-6-6 \\\\ &=- \, 3 \end{aligned}

As y=x^{2}-2x-6, when x=2,

\begin{aligned}y&=(2)^{2}-2(2)-6 \\\\ &=4-4-6 \\\\ &=- \, 6 \end{aligned}

Both are valid so the answer is correct.

When graphed, these two equations intersect at (2, \, – \, 6) and (3, \, – \, 3). So the solutions to the system of equations are:

Example 3: equation of a straight line and a circle (factor)

Use y=x+7 to substitute the value of y into the first equation:

(x+7)^{2}+x^{2}=29

x^{2}+7x+10=0 can be factored:

(x+5)(x+2)=0

x+2=0 so x=- \, 2

The two values of x are x=- \, 5 and x=- \, 2.

As x+7=y, when x=- \, 5, \, y=- \, 5+7=2.

As x+7=y, when x=- \, 2, \, y=- \, 2+7=5.

x=- \, 5, \, y=2 and x=- \, 2, \, y=5.

Let’s use the equation of the circle to check the answers.

Substitute x=- \, 5, \, y=2 into y^2+x^2=29\text{:}

2^{2}+(- \, 5)^{2}=4+25=29.

Substitute x=- \, 2, \, y=5 into y^2+x^2=29\text{:}

5^{2}+(- \, 2)^{2}=25+4=29.

Let’s look at the graph of the system:

When graphed these two equations intersect at (– \, 2, \, 5) and (- \, 5, \, 2), so the solutions to the system of equations are:

Notice that the graph of y^2+x^2=29 is a circle!

Example 4: equation of a straight line and a circle (quadratic formula)

Use x=y+2 to substitute the value of x into the first equation:

y^2+(y+2)^2=14

This equation does not factor, so use the quadratic formula to determine the value(s) of x\text{:}

y^{2}+2y-5=0 with a=1, \, b=2, \, c=- \, 5

\begin{aligned}y&=\cfrac{- \, b\pm\sqrt{b^{2}-4ac}}{2a} \\\\ &=\cfrac{- \, 2\pm\sqrt{2^{2}-4(1)(- \, 5)}}{2(1)} \\\\ &=\cfrac{- \, 2\pm\sqrt{4+20}}{2} \\\\ &=\cfrac{- \, 2\pm2\sqrt{6}}{2} \end{aligned}

y=- \, 1+\sqrt{6} or y=- \, 1-\sqrt{6}

The two values of y can be substituted into one of the original equations to find the two possible values of x.

As x=y+2, when y=-1+\sqrt{6}\text{:}

x=- \, 1+\sqrt{6}+2=1+\sqrt{6}

As x=y+2, when y=- \, 1-\sqrt{6}\text{:}

x=- \, 1-\sqrt{6}+2=1-\sqrt{6}

Substitute x=1+\sqrt{6},~y=- \, 1+\sqrt{6} into y^{2}+x^{2}=14\text{:}

\begin{aligned}&(- \, 1+\sqrt{6})^{2}+(1+\sqrt{6})^{2} \\\\ &=(- \, 1+\sqrt{6})(-1+\sqrt{6})+(1+\sqrt{6})(1+\sqrt{6}) \\\\ &=(1-\sqrt{6}-\sqrt{6}+6)+(1+\sqrt{6}+\sqrt{6}+6) \\\\ &=7-2\sqrt{6}+7+2\sqrt{6} \\\\ &=14. \end{aligned}

Substitute x=1-\sqrt{6},~y=- \, 1-\sqrt{6} into y^{2}+x^{2}=14\text{:}

\begin{aligned}&(- \, 1-\sqrt{6})^{2}+(1-\sqrt{6})^{2} \\\\ &=(- \, 1-\sqrt{6})(-1-\sqrt{6})+(1-\sqrt{6})(1-\sqrt{6}) \\\\ &=(1+\sqrt{6}+\sqrt{6}+6)+(1-\sqrt{6}-\sqrt{6}+6) \\\\ &=7+2\sqrt{6}+7-2\sqrt{6} \\\\ &=14.\end{aligned}

When graphed these two equations intersect at (1+\sqrt{6}, \, - \, 1+\sqrt{6}) and (1-\sqrt{6}, \, - \, 1-\sqrt{6}), so the solutions to the system of equations are:

x=1+\sqrt{6}, \, y=- \, 1+\sqrt{6} and x=1-\sqrt{6}, \, y=- \,1-\sqrt{6}

Notice that the graph of y^2+x^2=14 is a circle!

Example 5: equation of a straight line and a hyperbola (quadratic formula/factor)

Solve the system of equations:

First, make y the subject of the formula in the second equation.

\begin{aligned}3x+4y&=7 \\\\ 4y&=7-3x \\\\ y&=\cfrac{7-3x}{4} \end{aligned}

Then, use the fact that y=\cfrac{7-3x}{4} to substitute the value of y into the first equation:

2x^{2}-8\left(\cfrac{7-3x}{4}\right)^{2}=18.

\begin{aligned}2x^{2}-8\left(\cfrac{7-3x}{4}\right)^{2}&=18 \\\\ 2x^{2}-8\left(\cfrac{7-3x}{4}\right)\left(\cfrac{7-3x}{4}\right)&=18 \\\\ 2x^{2}-8\left(\cfrac{49-21x-21x+9x^{2}}{16}\right)&=18 \\\\ 2x^{2}-8\left(\cfrac{49-42x+9x^{2}}{16}\right)&=18 \\\\ 2x^{2}- \cfrac{49-42x+9x^{2}}{2}&=18 \\\\ 4x^{2}-\left(49-42x+9x^{2}\right)&=36 \\\\ 4x^{2}-49+42x-9x^{2}&=36 \\\\ - \, 49+42x-5x^{2}&=36 \\\\ 42x-5x^{2}&=85 \\\\ - \, 5x^{2}&=- \, 42x+85 \\\\ 0&=5x^{2}-42x+85 \end{aligned}

This equation appears harder to factor, so use the quadratic formula:

5x^{2}-42x+85=0 where a=5, \, b =- \, 42, \, c= 85

\begin{aligned}x&=\cfrac{- \, b\pm\sqrt{b^{2}-4ac}}{2a} \\\\ &=\cfrac{- \, (- \, 42)\pm\sqrt{(- \, 42)^{2}-4(5)(85)}}{2(5)} \\\\ &=\cfrac{42\pm\sqrt{1764-1700}}{10} \\\\ &=\cfrac{42\pm\sqrt{64}}{10} \\\\ &=\cfrac{42\pm{8}}{10} \end{aligned}

x=3.4 or x=5

Notice that the solutions can be decimals as well as integers.

Since there are two values of x, substitute both values into one of the original equations and find the two possible values of y.

As 3x+4y=7, when x=3.4,

\begin{aligned}3(3.4)+4y&=7 \\\\ 10.2+4y&=7 \\\\ 4y&=- \, 3.2 \\\\ y&=- \, 0.8 \end{aligned}

As 3x+4y=7, when x=5,

\begin{aligned}3(5)+4y&=7 \\\\ 15+4y&=7 \\\\ 4y&=- \, 8 \\\\ y&=- \, 2 \end{aligned}

x=3.4, \, y=- \, 0.8 and x=5, \, y=- \, 2.

As 2x^{2}-8y^{2}=18 where x=3.4, \, y=- \, 0.8,

2(3.4)^{2}-8(- \, 0.8)^{2}=23.12-5.12=18

As 2x^{2}-8y^{2}=18 where x=5, \, y=- \, 2,

2(5)^{2}-8(- \, 2)^{2}=50-32=18

Let’s look at part of the graph of the system.

When graphed, these two equations intersect at (3.4, \, – \,0.8) and (5, \, – \, 2), so the solutions to the system of equations are:

Example 6: equation of a straight line and an ellipse (quadratic formula)

\begin{aligned}7x+y&=28 \\\\ y&=28-7x \end{aligned}

Then, use the fact that y=28-7x to substitute the value of y into the first equation:

x^{2}+8\left(28-7x\right)^{2}=20.

\begin{aligned}x^{2}+8(28-7x)^{2}&=20 \\\\ x^{2}+8(28-7x)(28-7x)&=20 \\\\ x^{2}+8(784-196x-196x+49x^{2})&=20 \\\\ x^{2}+8(784-392x+49x^{2})&=20 \\\\ x^{2}+6272-3136x+392x^{2}&=20 \\\\ 393x^{2}-3136x+6272&=20 \\\\ 393x^{2}-3136x+6252=0 \end{aligned}

This equation would be very hard to factor, so use the quadratic formula:

393x^{2}-3136x+6252=0 where a=393, \, b=- \, 3136, \, c=6252

\begin{aligned}x&=\cfrac{- \, b\pm\sqrt{b^{2}-4ac}}{2a} \\\\ &=\cfrac{- \, (- \, 3136)\pm\sqrt{(- \, 3136)^{2}-4(393)(6252)}}{2(393)} \\\\ &=\cfrac{3136\pm\sqrt{9834496-9828144}}{786} \\\\ &=\cfrac{3136\pm\sqrt{6352}}{786} \\\\ &=\cfrac{3136\pm4\sqrt{397}}{786} \end{aligned}

x=4.091220656… or x=3.88842311…

As 7x+y=28, when x=4.091 \, (3dp),

\begin{aligned}7(4.091)+y&=28. \\\\ 28.637+y&=28 \\\\ y&=- \, 0.637 \end{aligned}

As 7x+y=28, when x=3.888 \, (3dp),

\begin{aligned}7(3.888)+y&=28. \\\\ 27.216+y&=28 \\\\ y&=0.784 \end{aligned}

x=4.091, \, y=- \, 0.637 and x=3.888, \, y=0.784.

As x^{2}+8y^{2}=20 where x=4.091, \, y=-0.637,

(4.091)^{2}+8(- \, 0.637)^{2}=16.736281+3.246152=20 (nearest integer)

As x^{2}+8y^{2}=20 where x=3.888, \, y=0.784,

(3.888)^{2}+8(0.784)^{2}=15.116544+4.917248=20 (nearest integer)

When graphed, these two equations intersect at (4.091, \, - \, 0.637) and (3.888, \, 0.784), so the solutions to the system of equations are:

Notice that the graph of x^{2}+8y^{2}=20 is an ellipse!

Teaching tips for nonlinear system of equations

• Explore systems of linear equations before beginning this topic. It is also a good idea to review skills like changing the subject, operating with square roots and expanding quadratic equations, before asking students to solve systems of nonlinear equations.
• Show students a variety of strategies, such as the substitution method or the elimination method, but never require students to solve in a certain way (unless necessary on an assessment). Letting students decide how to solve helps them understand the methods in their own time and allows them to develop analytical skills, such as thinking about which way to solve would be the most efficient.
• For students to truly be proficient at solving nonlinear systems, they need to see equations of all kinds, not just equations in standard form.

Easy mistakes to make

• Forgetting that squaring a negative number results in a positive The product of two negative numbers is always positive. It is also important to use parentheses when solving with a calculator to prevent this mistake. See how a calculator solves differently for each type of equation. For example, \begin{aligned}- \, 9^2&=- \, 81 \\\\ \left(- \, 9^2\right)&=81 \\\\ - \, \left(9^2\right)&=- \, 81 \end{aligned}
• Missing solutions It is easy to forget that a nonlinear system of equations can have two pairs of solutions. Use substitution to find all of the possible solutions.
• Substitute \textbf{y} or \textbf{x} incorrectly A term from the second equation should be isolated, before being used to substitute. Additionally, if the term isolated is y (as is common), then the expression it is equal to should be used as y in the first equation. Not doing this correctly will lead to mistakes. For example, The following system x^{2}-y^{2}=10 and y=4x+3 should be substituted in this way: x^{2}-(4x+3)^{2}=10
• Not subtracting the entire term within the parentheses If a term is being subtracted, all coefficients and constants in the term need to be subtracted. For example, 4x^2-\left(49-42x+9x^2\right)=4x^2-49+42x-9x^2
• Not simplifying algebraic fractions When using algebraic fractions to remove the denominator make sure each term is carefully multiplied. It is much easier to make a mistake when operations involve algebraic fractions. For example, \begin{aligned}&2x^{2}-\cfrac{49-42x+9x^{2}}{2} \\\\ &4x^{2}-(49-42x+9x^{2}) \end{aligned}

Related systems of equations lessons

• Systems of equations
• Intersecting lines

Practice nonlinear system of equations questions

1. Solve the nonlinear system of equations:

x=- \, 1, \, y=- \, 2 and x=- \, 3, \, y=- \, 2

x=1, \, y=2 and x=3, \, y=2

x=1, \, y=2 and x=- \, 3, \, y=- \, 2

x=1, \, y=- \, 2 and x=3, \, y=- \, 2

Use the fact that y=x+1 to substitute the value of y into the second equation: x+1=x^{2}+3x-2.

x^{2}+2x-3=0 can be factored.

x+3=0 so x=- \, 3

The two values of x can be substituted into one of the original equations and find the two possible values of y.

When x=- \, 3, \, \, y=- \, 3+1=- \, 2

When x=1, \, y=1+1=2

The two solutions are x=- \, 3, \, y=- \, 2 and x=1, \, y=2.

Note: Graph of the system for reference:

2. Solve the nonlinear system of equations:

x=- \, 3, \, y=0 and x=0, \, y=- \, 3

Use the fact that y=x+3 to substitute the value of y into the second equation: x+3=x^{2}+7x+12.

x^{2}+6x+9=0 can be factored.

x+3=0 so x=- \, 3 only.

The value of x can be substituted into one of the original equations and find the value of y. Note, in this case there is only one solution.

When x=- \, 3, \, y=- \, 3+3=0

The solution is x=- \, 3, \, y=0.

3. Solve the nonlinear system of equations:

x=5, \, y=9 and x=1, \, y=3

x=- \, 5, \, y=9 and x=1, \, y=3

x=- \, 5, \, y=9 and x=- \, 1, \, y=- \, 3

x=- \, 5, \, y=9 and x=1, \, y=- \, 3

Use the fact that y=4-x to substitute the value of y into the second equation: 4-x=x^{2}+3x-1.

x^{2}+4x-5=0 can be factored.

When x=- \, 5, \, – \, 5+y=4 so y=9

When x=1, \, 1+y=4 so y=3

The two solutions are x=- \, 5, \, y=9 and x=1, \, y=3.

4. Solve the nonlinear system of equations:

Use the fact that y=2+5x to substitute the value of y into the second equation: 2+5x=9x^2+11x+3.

This equations appears harder to factor, so use the quadratic formula:

9x^{2}+6x+1=0 where a=9, \, b=6, \, c=1

x=- \, \cfrac{1}{3} only

When x=- \, \cfrac{1}{3},

The one solution is x=- \, \cfrac{1}{3},~y=- \, \cfrac{11}{3}.

5. Solve the nonlinear system of equations:

x=1+2\sqrt{3},~y=3+2\sqrt{3} and x=- \, 1+2\sqrt{3},~y=- \, 3+2\sqrt{3}

x=- \, 1-2\sqrt{3},~y=- \, 3-2\sqrt{3} and x=1-2\sqrt{3},~y=3-2\sqrt{3}

x=1+2\sqrt{3},~y=- \, 3-2\sqrt{3} and x=- \, 1+2\sqrt{3},~y=3-2\sqrt{3}

x=- \, 1-2\sqrt{3},~y=- \, 3-2\sqrt{3} and x=- \, 1+2\sqrt{3},~y=- \, 3+2\sqrt{3}

Use the fact that   y= x-2 to substitute the value of y into the second equation: 2x^{2}-x(x-2)=11.

x^{2}+2x-11=0 with a=1, \, b=2, \, c=- \, 11

x=- \, 1+2\sqrt{3} or x=- \, 1-2\sqrt{3}

When x=- \, 1+2\sqrt{3},

y=- \, 1+2\sqrt{3}-2 so y=- \, 3+2\sqrt{3}

When x=- \, 1-2\sqrt{3},

y=- \, 1-2\sqrt{3}-2 so y=- \, 3-2\sqrt{3}

The two solutions are x=- \, 1+2\sqrt{3},~y=- \, 3+2\sqrt{3} and x=- \, 1-2\sqrt{3},~y=- \, 3-2\sqrt{3}.

6. Solve the nonlinear system of equations:

x=0.5,~y=- \, 0.75 and x=- \, 2,~y=- \, 2

x=- \, 0.5,~y=- \, 0.75 and x=2,~y=- \, 2

x=0.5,~y=0.75 and x=2,~y=2

x=0.5,~y=- \, 0.75 and x=2,~y=- \, 2

First, make y the subject of the formula in the first equation.

Use the fact that   \cfrac{x-2}{2} to substitute the value of y into the second equation: \cfrac{x-2}{2}=x^{2}+2x-2.

2x^{2}+3x-2=0 with a=2, \, b=3, \, c=- \, 2

x=\cfrac{- \, 3+5}{4}=\cfrac{1}{2} or x=\cfrac{- \, 3-5}{4}=- \, 2

When x=\cfrac{1}{2},

When x=- \, 2,

The two solutions are x=\cfrac{1}{2},~y=- \, \cfrac{3}{4} and x=- \, 2,~y=- \, 2.

Nonlinear systems of equations FAQs

Functions that create straight line graphs.

Yes, any equation can be part of a nonlinear system of equation.

A mathematical expression that has variables, coefficients and exponents.

The next lessons are

• Number patterns
• Functions in algebra
• Laws of exponents

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