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Expressions and Equations 6th Grade

Welcome to our Expressions and Equations 6th Grade Worksheets.

Here you will find a range of algebra worksheets to help you learn about basic algebra, including generating and calculating algebraic expressions and solving simple equations.

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Expressions and Equations Support

Generate the expression worksheets, calculate the expression worksheets.

  • Simplify Expressions Worksheets
  • Solve the Equations Worksheets
  • Expressions and Equations Mixed Worksheets
  • More recommended resources
  • Expressions and Equations 6th Grade Online Quiz

Want to gain a basic understanding of algebra?

Looking for some simple algebra worksheets?

Do you need a bank of useful algebra resources?

Look no further! The pages you need are below!

Here is our selection of basic algebra sheets to try.

We have split the worksheets up into 3 different sections:

  • 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.
  • Mixed questions involving all 3 of the above.

By splitting the algebra up into sections, you only need to concentrate on one aspect at a time!

Each question sheet comes with its own separate answer sheet.

Want to test yourself to see how well you have understood this skill?.

  • Try our NEW quick quiz at the bottom of this page.

What is an algebraic expression?

An expression is a mathematical statement where variables and operations are combined.

  • 2a + 5 is an expression involving the variable a
  • 5(y 2 - 6) is another expression

What is an algebraic equation?

An equation is where an algebraic expression is equal to something, which might be a number, or another algebraic expression.

  • 2a + 5 = 7 is an equation
  • 5(y 2 - 6) = 3y + 8 is another equation

How to Generate an Expression

When we are generating an expression, we are taking a rule and turning it into algebra.

  • Subtract 6 from n could be written as n - 6.
  • Multiply d by 4 could be written as d x 4 or 4d.
  • Add 5 to p and then double the result is written as (p + 5) x 2 or 2(p + 5)

How to Calculate an Expression

When we are calculating the value of an expression, we work out the value of the expression when we give a value to the variable.

  • p + 5 has a value of 11 when p = 6 because 6 + 5 = 11
  • 4 - q has a value of 1 ½ when q = 2 ½ because 4 - 2 ½ = 1 ½
  • 4(n - 2) has a value of 32 when n = 10 because 4 x (10 - 2) = 4 x 8 = 32
  • 4(n - 2) has a value of -8 when n = 0 because 4 x (0 - 2) = 4 x (-2) = -8

How to Solve a Simple Equation

When we are solving an equation, we are finding out the value(s) of the variable in the equation.

  • then p = 4 because 4 + 5 = 9
  • Answer: p = 4
  • then n = 56 ÷ 7 = 8
  • Answer: n = 8
  • means that f = 4 because 12 - 4 = 8
  • Answer: f = 4
  • then t = 5 because 0.6 x 5 = 3
  • Answer: t = 5
  • then r = 32 because ½ x 32 = 16
  • Answer: r = 32

Expressions and Equations 6th Grade Worksheets

  • Generate the Expressions Sheet 6:1
  • PDF version
  • Generate the Expressions Sheet 6:2
  • Calculate the Expression Sheet 6:1
  • Calculate the Expression Sheet 6:2

Calculate the Expressions Walkthrough Video

This short video walkthrough shows several problems from our Calculate the Expression Worksheet 6:1 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, please check out the video below!

Simplifying Expressions Worksheets

  • Simplifying Expressions Sheet 6:1
  • Simplifying Expressions Sheet 6:2

Solve the Equation Worksheets

  • Solving Equations Sheet 6:1
  • Solving Equations Sheet 6:2

Expressions and Equations 6th Grade Mixed Worksheets

These questions involve solving equations, working out the value of expressions and also generating expressions.

They are a combination of questions from all the above categories.

  • Expressions and Equations Mixed Questions Sheet 6:1
  • Expressions and Equations Mixed Questions Sheet 6:2

Expressions and Equations Walkthrough Video

This short video walkthrough shows several problems from our Expressions and Equations Mixed Questions Worksheet 6:1 being solved and has been produced by the West Explains Best math channel.

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Inequalities Worksheets

  • Inequalities on a Number Line
  • Writing Inequalities from Word Problems
  • 6th Grade Distributive Property Worksheets

The sheets on this page have been designed to factorize and expand a range of simple expressions using the distributive property..

  • Basic Algebra Worksheets

We have a selection of basic algebra worksheets which are aimed at 6th and 7th graders and involve similar skills to the sheets here.

Free Algebra Problem Solver

The Mathway Calculator is a great way to solve algebra problems that you can type into a calculator.

Try using this online calculator tool to solve one of your problems and watch it work!

There are a range of calculators to choose from to meet your needs.

The Mathway problem solver will answer your problem instantly and also give you a link to view each of the steps needed.

If you choose to 'View the steps' you will be directed to the Mathway website where you will be able to see in more detail each of the steps needed to solve the problem. Please note that Mathway may charge you a small fee for this!

Factorising Quadratic Equations

Are you stuck on a quadratic equation and don't know what to do?

Are you looking for some worksheets on factorising quadratic equations to print out?

Take a look at our support pages on quadratic equations where you will hopefully find what you are looking for.

  • Factorising Quadratic Equations Support page
  • Factoring Quadratic Equations worksheets
  • Algebra Math Games

If you are looking for a fun printable algebra game to play then try out our algebra game page.

You will find a range of algebra games that make learning algebra fun and non-threatening.

The only equipment you need is a scientific calculator, some dice, and a few counters!

PEMDAS Worksheets

The sheets in this section involve using parentheses and exponents in simple calculations.

There are also lots of worksheets designed to practice and learn about PEMDAS.

Using these worksheets will help your child to:

  • know and understand how parentheses works;
  • understand how exponents work in simple calculations.
  • understand and use PEMDAS to solve a range of problems.
  • PEMDAS Problems Worksheets 5th Grade
  • 6th Grade Order of Operations

Interactive Equality Explorer

This interactive equality explorer has been produced by PhET Interactive Simulations at the University of Colorado.

It is a useful tool for exploring different ideas including negative numbers and algebra equations and equality.

Probably the most useful part of the app is to use the 'Solve It' section once you are confident how it works.

You can then select your level of difficulty and start solving some algebraic equations by getting your variables onto one side of the equation and the numerical values on the other, and then multiplying or dividing the equation until you find the value of the required variable.

interactive equality explorer by PhET

  • Interactive Equality Explorer by PhET

Expressions and Equations 6th Grade Quiz

Our quizzes have been created using Google Forms.

At the end of the quiz, you will get the chance to see your results by clicking 'See Score'.

This will take you to a new webpage where your results will be shown. You can print a copy of your results from this page, either as a pdf or as a paper copy.

For incorrect responses, we have added some helpful learning points to explain which answer was correct and why.

We do not collect any personal data from our quizzes, except in the 'First Name' and 'Group/Class' fields which are both optional and only used for teachers to identify students within their educational setting.

We also collect the results from the quizzes which we use to help us to develop our resources and give us insight into future resources to create.

For more information on the information we collect, please take a look at our Privacy Policy

We would be grateful for any feedback on our quizzes, please let us know using our Contact Us link, or use the Facebook Comments form at the bottom of the page.

This quick expressions and equations 6th grade quiz tests your knowledge and skill at generating and calculating expressions, as well as solving equations.

How to Print or Save these sheets 🖶

Need help with printing or saving? Follow these 3 steps to get your worksheets printed perfectly!

  • How to Print support

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solving algebraic equations year 6

For this sequence, the first five terms would be 5, 7, 9, 11, 13.

Express missing number problems algebraically

Your child will use letters to represent unknown numbers when solving number problems. For example:

Instead of  writing ? + 3 = 10 , you child could write  y + 3 = 10 .

Find pairs of numbers that satisfy an equation with 2 unknown numbers

Your child will be able to find multiple solutions to equations with two unknown numbers. For example:

a + b = 8. What could  a and b be?   The positive values for a and b could be: 7 and 1, 1 and 7, 6 and 2, 2 and 6, 5 and 3, or 3 and 5. (As the two unknown values are represented by different letters, we will assume at this stage that a and b must be different numbers, therefore the value of a and b could not be 4 and 4.)
2 a + b = 28. What could a and b be?   Your child will use trial and error to find different possibilities. If they take the value of a to be 4, then 2 a = 8. So, 8 + b = 28. Therefore, b = 28 – 8. So, b = 20. a = 4 and b = 20 is therefore one answer to this equation.

How to help at home

There are lots of everyday ways you can help your child to understand algebra. Here are just a few ideas.

1. Practise basic algebra

Your child will have solved lots of problems involving missing numbers at school. Before Year 6, the unknown number in a calculation will have been represented using a blank box or a question mark. This will now be replaced by a letter, like a or b . This letter represents the unknown number, also known as the variable .

There are lots of ways you could help your child solve problems where there are one or more variables. For instance, why not play number puzzles such as the one below?

solving algebraic equations year 6

Each shape has a different value. The total value of the shapes in each column and row is shown at the end of the column or row. See if your child can work out the value of each shape and then work out the missing totals.

2. Play with sequences

Below are a few steps you can take to help your child get to know linear sequences:

  • Choose a sequence of five numbers. Try to begin with sequences of numbers in the times tables. For example, start with 3 and write down the next four next terms in the sequence: 3, 6, 9, 12, 15.
  • Can your child describe the number sequence? What is happening to the numbers in the sequence? In our example, the numbers are increasing by 3 each time, so there is a difference of 3 between each of the terms.
  • Ask your child to predict the next few numbers. They should see that they just need to add three to get the next term. Therefore, the next two numbers in the sequence would be 18 (15 + 3) and then 21 (18 + 3).
  • Can your child predict what the tenth number in the sequence would be? They could do this by adding on 3 ten times to reach 30.
  • Encourage your child to look at the relationship between the position of each term (for example, the 3rd number in the sequence) and the value of that term (for example, 9). They could make a table to help them identify patterns and find a general rule:

solving algebraic equations year 6

Because the number in the sequence is always the term multiplied by 3, this sequence can be written as the algebraic rule 3 n .

Here are some ideas for an extra challenge once your child has followed those steps:

  • Ask your child to work out the hundredth number in the sequence. Your child should be able to see that if we multiply the term number by 3, we will get the sequence number. For example, the 100th term would be 300 because 3 × 100 = 300.
  • Ask your child if they could find any number in the sequence, like the 736th term.
  • Choose a number that you know is not in the sequence (for example, 37) and ask your child to figure out if this number would appear in the sequence. How do they know?
  • You could ask your child to explore number sequences where the numbers decrease by a constant quantity (for example, 40, 36, 32, 28, and so on).

Activity: Linear sequences

solving algebraic equations year 6

Practise your algebra skills with these questions about linear sequences.

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Year 6 Algebra Eight Superb Teaching Tips!

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National Curriculum – Year 6 Algebra

In your students final year of primary education, a new topic year 6 algebra is introduced. Let’s make that introduction a little easier with these top tips for teaching algebra!

Top Tip 1 - Using Visual Representations

Creating visual representations of the problem being solved can help students to understand what algebra actually is and what steps they need to take to begin solving the question.

The best part about bar models is that they can be adapted for both simple and more complex questions, thus can be used for all levels of understanding when teaching year 6 algebra!

Example Visual: Bar Models

This bar model represents the simple equation 2x = 10 . Using the bar model, students can clearly see that to solve this they would need to divide 10 by 2 to find x.

As you can see from the example above, bar models can help students visualise the problem they are solving, making it easier for them to work out.  Getting students to create the bar models on their own from the question is also an effective way of gaining their understanding of year 6 algebra! 

Bar models are a great way of helping students get used to using simple formulae and enables them to express missing number problems algebraically.

Top Tip 2 - Using Shapes!

Using shapes to represent values in a year 6 algebra problem helps students to begin thinking algebraically, without the initial confusion of letters and numbers. 

Using this example, students would be required to work out the value of the red triangle in each sum. This allows them to begin thinking algebraically , as they are working out an unknown value (which would normally be represented as a letter) from the values which they have already been given.  This is a fun way of getting students to express missing number problems algebraically. This method of using shapes to represent values can be increased in complexity by having two unknown values instead of one. These would have to be kept the same throughout in order for the students to work it out.

Top Tip 3 - Arithmetic Sequence Starters

Using easy linear number sequences as a starter activity will help students gain some confidence before getting into some of the trickier algebra questions. The example to the right would require students to work out the next term in the sequence. To increase the difficulty of this task, students would be asked to find the nth term of each sequence.

As shown in the example above, students will be acquiring the skill to generate and describe linear sequences, which is a requirement of the national curriculum for year 6 algebra. This is a key base skill to learning and understanding algebra.

Top Tip 4 - Group Work!

Getting students to work together on solving equations can motivate them to work harder and encourage them to keep trying! A good group activity is the hexagon calculation game, which requires students to roll a dice and whatever number is shown on the dice is the value of n. The student that rolled the dice gets to choose a hexagon (which contains an algebra problem) to solve. The student that solves the most hexagons wins! 

The example below shows what your students would be doing:

As you can see from the example above, your students will be working through a range of algebra problems with varying difficulty. This challenge allows pupils to work together and have fun with year 6 algebra!

Top Tip 5 - Show Every Step!

When learning year 6 algebra it’s important for students to see every single step involved in solving the problem. Teaching students to write down each step in their books and show you their working out allows them to clearly see the process that they’re being taught. 

An excellent benefit of this (especially as it gets more difficult) is that you can see in which stage of the question a student may have gone wrong. This means that you can identify the issue and help the student accordingly. 

As the teacher, you could demonstrate this by going through example questions on the whiteboard. This way you can clearly show the class the correct way to work out more complex problems. Another way to encourage them to show step by step working out is to award extra marks for each step!

As shown in the example above, each step is written down. The pupil shows all their working out, making it easy to mark and identify any potential mistakes!

Top Tip 6 - Use Engaging Videos

Using educational videos is a fun and easy way of explaining topics to students. If some students are struggling to understand algebra, having someone else explain it in a different way is often an effective method of increasing their understanding. Sometimes hearing it explained it a new way is what helps the topic click!

The video above is a really good introduction to algebra and goes over all the fundamentals that are needed in year 6 algebra! Videos like these are perfect for engaging students in a more relaxed and fun way.

Top Tip 7 - Using Number Riddles

Number riddles are a fabulous way of making year 6 algebra fun, whilst also keeping students engaged! The prospect of solving the riddle keeps pupils motivated when doing algebra (which is normally an unpopular topic). Using fun activities like this encourages students to view algebra in a more positive way, thus helping them learn.

As you can see from the example above, using riddles is an enjoyable method for getting students to engage with algebra’s trickier content. Solving both an algebra equation and a riddle is a fun challenge for pupils. This is a versatile activity and can be used in many different topics. If students enjoy it, it’s an excellent activity to implement into your lessons! A great site for more year 6 algebra activities is  NCETM.

Top Tip 8 - Using Emile!

Emile is a game-based learning app for schools and homes. Emile has an engaging story that students will LOVE! One example of the games made for learning year 6 algebra is the Rating game, which allows students to go through questions at their own pace and unlock new clothes for their tamagotchi. 

Each student has their own tamagotchi which they can feed, play with and grow as they unlock more levels. This encourages and motivates students to continue through their learning in a fun way! Each student will be automatically allocated the right work, ensuring their progress!

To the left you can see another example of what Emile has to offer when learning and teaching year 6 algebra. By signing up for a demo, you can see what Emile has to offer you and how it can help your pupils achieve more!

Printable Year 6 Algebra Worksheets:

Riddle Me This Worksheet

Hexagon Game Worksheet

Linear Number Sequences Worksheet

Using Shapes Worksheet

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Solving Equations

What is an equation.

An equation says that two things are equal. It will have an equals sign "=" like this:

That equations says:

what is on the left (x − 2)  equals  what is on the right (4)

So an equation is like a statement " this equals that "

What is a Solution?

A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .

Example: x − 2 = 4

When we put 6 in place of x we get:

which is true

So x = 6 is a solution.

How about other values for x ?

  • For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
  • For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

More Than One Solution

There can be more than one solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3 , or x = 2

When we gather all solutions together it is called a Solution Set

The above solution set is: {2, 3}

Solutions Everywhere!

Some equations are true for all allowed values and are then called Identities

Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities

Let's try θ = 30°:

sin(−30°) = −0.5 and

−sin(30°) = −0.5

So it is true for θ = 30°

Let's try θ = 90°:

sin(−90°) = −1 and

−sin(90°) = −1

So it is also true for θ = 90°

Is it true for all values of θ ? Try some values for yourself!

How to Solve an Equation

There is no "one perfect way" to solve all equations.

A Useful Goal

But we often get success when our goal is to end up with:

x = something

In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.

Example: Solve 3x−6 = 9

Now we have x = something ,

and a short calculation reveals that x = 5

Like a Puzzle

In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.

Here are some things we can do:

  • Add or Subtract the same value from both sides
  • Clear out any fractions by Multiplying every term by the bottom parts
  • Divide every term by the same nonzero value
  • Combine Like Terms
  • Expanding (the opposite of factoring) may also help
  • Recognizing a pattern, such as the difference of squares
  • Sometimes we can apply a function to both sides (e.g. square both sides)

Example: Solve √(x/2) = 3

And the more "tricks" and techniques you learn the better you will get.

Special Equations

There are special ways of solving some types of equations. Learn how to ...

  • solve Quadratic Equations
  • solve Radical Equations
  • solve Equations with Sine, Cosine and Tangent

Check Your Solutions

You should always check that your "solution" really is a solution.

How To Check

Take the solution(s) and put them in the original equation to see if they really work.

Example: solve for x:

2x x − 3 + 3 = 6 x − 3     (x≠3)

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x − 3) :

2x + 3(x−3) = 6

Bring the 6 to the left:

2x + 3(x−3) − 6 = 0

Expand and solve:

2x + 3x − 9 − 6 = 0

5x − 15 = 0

5(x − 3) = 0

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

Hang On: 3 − 3 = 0 That means dividing by Zero!

And anyway, we said at the top that x≠3 , so ...

x = 3 does not actually work, and so:

There is No Solution!

That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!

This gives us a moral lesson:

"Solving" only gives us possible solutions, they need to be checked!

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Solving Equations Worksheets

Solving Equations Worksheets

Subject: Mathematics

Age range: 11-14

Resource type: Worksheet/Activity


Last updated

16 December 2014

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Solving equations

Here you will learn about solving equations, including linear and quadratic algebraic equations, and how to solve them.

Students will first learn about solving equations in grade 8 as a part of expressions and equations, and again in high school as a part of reasoning with equations and inequalities.

What is solving an equation?

Solving equations is a step-by-step process to find the value of the variable. A variable is the unknown part of an equation, either on the left or right side of the equals sign. Sometimes, you need to solve multi-step equations which contain algebraic expressions.

To do this, you must use the order of operations, which is a systematic approach to equation solving. When you use the order of operations, you first solve any part of an equation located within parentheses. An equation is a mathematical expression that contains an equals sign.

For example,

\begin{aligned}y+6&=11\\\\ 3(x-3)&=12\\\\ \cfrac{2x+2}{4}&=\cfrac{x-3}{3}\\\\ 2x^{2}+3&x-2=0\end{aligned}

There are two sides to an equation, with the left side being equal to the right side. Equations will often involve algebra and contain unknowns, or variables, which you often represent with letters such as x or y.

You can solve simple equations and more complicated equations to work out the value of these unknowns. They could involve fractions, decimals or integers.

What is solving an equation?

Common Core State Standards

How does this relate to 8 th grade and high school math?

  • Grade 8 – Expressions and Equations (8.EE.C.7) Solve linear equations in one variable.
  • High school – Reasoning with Equations and Inequalities (HSA.REI.B.3) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of math equations. 10+ questions with answers covering a range of 6th, 7th and 8th grade math equations topics to identify areas of strength and support!

How to solve equations

In order to solve equations, you need to work out the value of the unknown variable by adding, subtracting, multiplying or dividing both sides of the equation by the same value.

  • Combine like terms .
  • Simplify the equation by using the opposite operation to both sides.
  • Isolate the variable on one side of the equation.

Solving equations examples

Example 1: solve equations involving like terms.

Solve for x.

Combine like terms.

Combine the q terms on the left side of the equation. To do this, subtract 4q from both sides.

The goal is to simplify the equation by combining like terms. Subtracting 4q from both sides helps achieve this.

After you combine like terms, you are left with q=9-4q.

2 Simplify the equation by using the opposite operation on both sides.

Add 4q to both sides to isolate q to one side of the equation.

The objective is to have all the q terms on one side. Adding 4q to both sides accomplishes this.

After you move the variable to one side of the equation, you are left with 5q=9.

3 Isolate the variable on one side of the equation.

Divide both sides of the equation by 5 to solve for q.

Dividing by 5 allows you to isolate q to one side of the equation in order to find the solution. After dividing both sides of the equation by 5, you are left with q=1 \cfrac{4}{5} \, .

Example 2: solve equations with variables on both sides

Combine the v terms on the same side of the equation. To do this, add 8v to both sides.


After combining like terms, you are left with the equation 15v=8.

Simplify the equation by using the opposite operation on both sides and isolate the variable to one side.

Divide both sides of the equation by 15 to solve for v. This step will isolate v to one side of the equation and allow you to solve.

15v \div 15=8 \div 15

The final solution to the equation 7v=8-8v is \cfrac{8}{15} \, .

Example 3: solve equations with the distributive property

Combine like terms by using the distributive property.

The 3 outside the parentheses needs to be multiplied by both terms inside the parentheses. This is called the distributive property.

\begin{aligned}& 3 \times c=3 c \\\\ & 3 \times(-5)=-15 \\\\ &3 c-15-4=2\end{aligned}

Once the 3 is distributed on the left side, rewrite the equation and combine like terms. In this case, the like terms are the constants on the left, –15 and –4. Subtract –4 from –15 to get –19.

Simplify the equation by using the opposite operation on both sides.

The goal is to isolate the variable, c, on one side of the equation. By adding 19 to both sides, you move the constant term to the other side.

\begin{aligned}& 3 c-19+19=2+19 \\\\ & 3 c=21\end{aligned}

Isolate the variable to one side of the equation.

To solve for c, you want to get c by itself.

Dividing both sides by 3 accomplishes this.

On the left side, \cfrac{3c}{3} simplifies to c, and on the right, \cfrac{21}{3} simplifies to 7.

The final solution is c=7.

As an additional step, you can plug 7 back into the original equation to check your work.

Example 4: solve linear equations

Combine like terms by simplifying.

Using steps to solve, you know that the goal is to isolate x to one side of the equation. In order to do this, you must begin by subtracting from both sides of the equation.

\begin{aligned} & 2x+5=15 \\\\ & 2x+5-5=15-5 \\\\ & 2x=10 \end{aligned}

Continue to simplify the equation by using the opposite operation on both sides.

Continuing with steps to solve, you must divide both sides of the equation by 2 to isolate x to one side.

\begin{aligned} & 2x \div 2=10 \div 2 \\\\ & x= 5 \end{aligned}

Isolate the variable to one side of the equation and check your work.

Plugging in 5 for x in the original equation and making sure both sides are equal is an easy way to check your work. If the equation is not equal, you must check your steps.

\begin{aligned}& 2(5)+5=15 \\\\ & 10+5=15 \\\\ & 15=15\end{aligned}

Example 5: solve equations by factoring

Solve the following equation by factoring.

Combine like terms by factoring the equation by grouping.

Multiply the coefficient of the quadratic term by the constant term.

2 x (-20) = -40

Look for two numbers that multiply to give you –40 and add up to the coefficient of 3. In this case, the numbers are 8 and –5 because 8 x -5=–40, and 8+–5=3.

Split the middle term using those two numbers, 8 and –5. Rewrite the middle term using the numbers 8 and –5.


Group the terms in pairs and factor out the common factors.

2x^2+8x-5x-20=2x(x + 4)-5(x+4)=0

Now, you’ve factored the equation and are left with the following simpler equations 2x-5 and x+4.

This step relies on understanding the zero product property, which states that if two numbers multiply to give zero, then at least one of those numbers must equal zero.

Let’s relate this back to the factored equation (2x-5)(x+4)=0

Because of this property, either (2x-5)=0 or (x+4)=0

Isolate the variable for each equation and solve.

When solving these simpler equations, remember that you must apply each step to both sides of the equation to maintain balance.

\begin{aligned}& 2 x-5=0 \\\\ & 2 x-5+5=0+5 \\\\ & 2 x=5 \\\\ & 2 x \div 2=5 \div 2 \\\\ & x=\cfrac{5}{2} \end{aligned}

\begin{aligned}& x+4=0 \\\\ & x+4-4=0-4 \\\\ & x=-4\end{aligned}

The solution to this equation is x=\cfrac{5}{2} and x=-4.

Example 6: solve quadratic equations

Solve the following quadratic equation.

Combine like terms by factoring the quadratic equation when terms are isolated to one side.

To factorize a quadratic expression like this, you need to find two numbers that multiply to give -5 (the constant term) and add to give +2 (the coefficient of the x term).

The two numbers that satisfy this are -1 and +5.

So you can split the middle term 2x into -1x+5x: x^2-1x+5x-5-1x+5x

Now you can take out common factors x(x-1)+5(x-1).

And since you have a common factor of (x-1), you can simplify to (x+5)(x-1).

The numbers -1 and 5 allow you to split the middle term into two terms that give you common factors, allowing you to simplify into the form (x+5)(x-1).

Let’s relate this back to the factored equation (x+5)(x-1)=0.

Because of this property, either (x+5)=0 or (x-1)=0.

Now, you can solve the simple equations resulting from the zero product property.

\begin{aligned}& x+5=0 \\\\ & x+5-5=0-5 \\\\ & x=-5 \\\\\\ & x-1=0 \\\\ & x-1+1=0+1 \\\\ & x=1\end{aligned}

The solutions to this quadratic equation are x=1 and x=-5.

Teaching tips for solving equations

  • Use physical manipulatives like balance scales as a visual aid. Show how you need to keep both sides of the equation balanced, like a scale. Add or subtract the same thing from both sides to keep it balanced when solving. Use this method to practice various types of equations.
  • Emphasize the importance of undoing steps to isolate the variable. If you are solving for x and 3 is added to x, subtracting 3 undoes that step and isolates the variable x.
  • Relate equations to real-world, relevant examples for students. For example, word problems about tickets for sports games, cell phone plans, pizza parties, etc. can make the concepts click better.
  • Allow time for peer teaching and collaborative problem solving. Having students explain concepts to each other, work through examples on whiteboards, etc. reinforces the process and allows peers to ask clarifying questions. This type of scaffolding would be beneficial for all students, especially English-Language Learners. Provide supervision and feedback during the peer interactions.

Easy mistakes to make

  • Forgetting to distribute or combine like terms One common mistake is neglecting to distribute a number across parentheses or combine like terms before isolating the variable. This error can lead to an incorrect simplified form of the equation.
  • Misapplying the distributive property Incorrectly distributing a number across terms inside parentheses can result in errors. Students may forget to multiply each term within the parentheses by the distributing number, leading to an inaccurate equation.
  • Failing to perform the same operation on both sides It’s crucial to perform the same operation on both sides of the equation to maintain balance. Forgetting this can result in an imbalanced equation and incorrect solutions.
  • Making calculation errors Simple arithmetic mistakes, such as addition, subtraction, multiplication, or division errors, can occur during the solution process. Checking calculations is essential to avoid errors that may propagate through the steps.
  • Ignoring fractions or misapplying operations When fractions are involved, students may forget to multiply or divide by the common denominator to eliminate them. Misapplying operations on fractions can lead to incorrect solutions or complications in the final answer.

Related math equations lessons

  • Math equations
  • Rearranging equations
  • How to find the equation of a line
  • Solve equations with fractions
  • Linear equations
  • Writing linear equations
  • Substitution
  • Identity math
  • One step equation

Practice solving equations questions

1. Solve 4x-2=14.

GCSE Quiz False

Add 2 to both sides.

Divide both sides by 4.

2. Solve 3x-8=x+6.

Add 8 to both sides.

Subtract x from both sides.

Divide both sides by 2.

3. Solve 3(x+3)=2(x-2).

Expanding the parentheses.

Subtract 9 from both sides.

Subtract 2x from both sides.

4. Solve \cfrac{2 x+2}{3}=\cfrac{x-3}{2}.

Multiply by 6 (the lowest common denominator) and simplify.

Expand the parentheses.

Subtract 4 from both sides.

Subtract 3x from both sides.

5. Solve \cfrac{3 x^{2}}{2}=24.

Multiply both sides by 2.

Divide both sides by 3.

Square root both sides.

6. Solve by factoring:

Use factoring to find simpler equations.

Set each set of parentheses equal to zero and solve.

x=3 or x=10

Solving equations FAQs

The first step in solving a simple linear equation is to simplify both sides by combining like terms. This involves adding or subtracting terms to isolate the variable on one side of the equation.

Performing the same operation on both sides of the equation maintains the equality. This ensures that any change made to one side is also made to the other, keeping the equation balanced and preserving the solutions.

To handle variables on both sides of the equation, start by combining like terms on each side. Then, move all terms involving the variable to one side by adding or subtracting, and simplify to isolate the variable. Finally, perform any necessary operations to solve for the variable.

To deal with fractions in an equation, aim to eliminate them by multiplying both sides of the equation by the least common denominator. This helps simplify the equation and make it easier to isolate the variable. Afterward, proceed with the regular steps of solving the equation.

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  • Inequalities
  • Types of graph
  • Coordinate plane

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