# Solve Equations with Unknown on Both Sides

In this worksheet, students will learn how to solve linear equations with variables on both sides of the equals sign by grouping the terms.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Equations

Difficulty level:

### QUESTION 1 of 10

'If I think of a number, multiply it by 10 and then add 2 it is the same as if I multiply it by 7 and add 14. What number am I thinking of?'

You may have seen puzzles like this before in your other work with equations, but this example is more complicated.

This time the answer is not just a number, we have to do something else to find the variable.

This type of puzzle can still be written as an equation, but now we have to use an x (or whatever letter we use to stand for the number) on each side of the equal sign.

So, the equation for this puzzle looks like this:

10x + 2 = 7x + 14

This looks more complicated, but it can still be solved using the balancing method, but we just have to add an extra step.

The first step is to change it into an equation where the elements involving x is only on one side.

There are 10x on the left and 7x on the right.

If we remove the 7x from the right, there will be no x's there.

But, remember whatever we do to the right, we must do to the left as well.

This is how we set this equation out:

Now, the equation should be simpler to solve.

We need to subtract 2 from both sides, then divide by 3.

Our full working will look like this:

Hopefully you're following this.

Let's try another example to check.

e.g. 3x - 5 = 10 - 2x

Now this time you will notice that on the right-hand side we have 2x, which is subtracted from 10.

So, what we do in this case is add 2x to both sides first to get rid of the -2x.

Here is our full working:

The rule to remember with these equations is to look for the side with the fewest x's on and take this from both sides.

Have a go at the questions and don't worry if you get them wrong, just look at the solution and remember to record all your working so you can compare.

In this activity, we will solve equations with variables on both sides of the equals sign by grouping the terms.

Consider the equation below:

7x - 3 = 4x + 12

Which of the options below is the best first step to take to solve this equation?

Subtract 4x from both sides

Subtract 7x from both sides

Consider the equation below:

2p - 8 = 12 - 3p

Which of the options below is the best first step to take to solve this equation?

Subtract 2p from both sides

Subtract 3p from both sides

There are three steps to solving the following equation:

8x - 22 = 2x + 8

The steps are listed below.

Can you put them in the correct order?

## Column B

1st
2nd
Subtract 2x from both sides
3rd
Divide both sides by 6

There are three steps to solving the following equation:

23 - 7y = 35 - 13y

The steps are listed below.

Can you put them in the correct order? ​

## Column B

1st
Divide both sides by 6
2nd
3rd
Subtract 23 from both sides

Which of the options below is the correct solution to this equation?

2x + 3 = 5x - 2

x = 3/5

x = 5/3

x = 2

x = 5

Which of the options below is the correct solution to this equation?

20 - 7x = 6x - 6

x = 1

x = 6/7

x = 7/6

x = 2

Match the equations below to their solutions.

## Column B

7a - 7 = 4a + 8
a = 3
8a - 24 = 12 - 4a
a = -2/7
6 - 4a = 10 - 5a
a = 5
3a + 6 = 4 - 4a
a = 4

This equation has brackets on the right-hand side as well as x's on both sides:

4(x - 2) = 3x + 5

Expand (multiply out) the brackets, then solve as usual.

## Column B

7a - 7 = 4a + 8
a = 3
8a - 24 = 12 - 4a
a = -2/7
6 - 4a = 10 - 5a
a = 5
3a + 6 = 4 - 4a
a = 4

This equation has brackets on both sides as well as variables on both sides:

5(2 - p) = 2(4 - 2p)

Expand both sets of brackets, then solve as usual.

## Column B

7a - 7 = 4a + 8
a = 3
8a - 24 = 12 - 4a
a = -2/7
6 - 4a = 10 - 5a
a = 5
3a + 6 = 4 - 4a
a = 4

Solve the following equation by expanding the brackets first:

3(4y + 3) = 2(3y + 5)

Type your answer below in the form a/b with no spaces, and using the / key to create your fraction bar.

## Column B

7a - 7 = 4a + 8
a = 3
8a - 24 = 12 - 4a
a = -2/7
6 - 4a = 10 - 5a
a = 5
3a + 6 = 4 - 4a
a = 4
• Question 1

Consider the equation below:

7x - 3 = 4x + 12

Which of the options below is the best first step to take to solve this equation?

Subtract 4x from both sides
EDDIE SAYS
The smallest number of x's is 4x on the right-hand side. Therefore, the best first step is to subtract 4x from both sides. Does that make sense?
• Question 2

Consider the equation below:

2p - 8 = 12 - 3p

Which of the options below is the best first step to take to solve this equation?

EDDIE SAYS
The smallest number of p's is -3p on the right-hand side. Remember that -3p is smaller than 2p. So the best thing for us to do first would be to add 2p to both sides. Did you remember to add rather than subtract, because the sign in front of the 3p is negative?
• Question 3

There are three steps to solving the following equation:

8x - 22 = 2x + 8

The steps are listed below.

Can you put them in the correct order?

## Column B

1st
Subtract 2x from both sides
2nd
3rd
Divide both sides by 6
EDDIE SAYS
The smallest number of x's is 2x on the right-hand side, so we need to get rid of this first. We do this by subtracting 2x from both sides. This leaves us with: 6x - 22 = 8 So, the next step is to add 22 to both sides. This leaves us with: 6x = 30 Finally, we have to divide by 6 leaving us with the answer: x = 5 Did you get those steps in the right order?
• Question 4

There are three steps to solving the following equation:

23 - 7y = 35 - 13y

The steps are listed below.

Can you put them in the correct order? ​

## Column B

1st
2nd
Subtract 23 from both sides
3rd
Divide both sides by 6
EDDIE SAYS
The smallest number of y's is -13y on the right-hand side. Since this number is negative, we need to add 13y to both sides first. This leaves us with: 23 + 6y = 35 The next step is to subtract 23 from both sides leaving us with: 6y = 12 Finally, we divide by 6 leaving us with an answer of: y = 2 Right, now to try solving some equations all the way through yourself.
• Question 5

Which of the options below is the correct solution to this equation?

2x + 3 = 5x - 2

x = 5/3
EDDIE SAYS
The smallest number of x's is on the left-hand side, so we need to subtract 2x from both sides to start. Your working should look something like this: 2x + 3 = 5x - 2 2x - 2x + 3 = 5x - 2x - 2 3 = 3x - 2 3 + 2 = 3x - 2 + 2 5 = 3x 5 ÷ 3 = 3x ÷ 3 5/3 = x Now since 3 does not divide into 5 exactly, this answer is best left as a top-heavy fraction. Do not convert it to a decimal with a calculator. In the exam, you will get full marks for a top-heavy fraction.
• Question 6

Which of the options below is the correct solution to this equation?

20 - 7x = 6x - 6

x = 2
EDDIE SAYS
Remember when there is a '-' in front of the x, we need to add it on, rather than subtract it. Here's the full working for this one: 20 - 7x = 6x - 6 20 - 7x + 7x = 6x + 7x - 6 20 = 13x - 6 20 + 6 = 13x - 6 + 6 26 = 13x 26 ÷ 13 = 13x ÷ 13 2 = x Are you getting the hang of these now? There's just a few more to solve!
• Question 7

Match the equations below to their solutions.

## Column B

7a - 7 = 4a + 8
a = 5
8a - 24 = 12 - 4a
a = 3
6 - 4a = 10 - 5a
a = 4
3a + 6 = 4 - 4a
a = -2/7
EDDIE SAYS
There are 4 equations to solve here be sure to take your time and solve one at a time. Your working should look something like this: 7a - 7 = 4a + 8 → -4a, then + 7, then ÷ 3 → a = 5 8a - 24 = 12 - 4a → + 4a, then + 24, then ÷ 12 → a = 3 6 - 4a = 10 - 5a → + 5a, then - 6 → a = 4 3a + 6 = 4 - 4a → + 4a, then - 6, then ÷ 7 → a = -2/7
• Question 8

This equation has brackets on the right-hand side as well as x's on both sides:

4(x - 2) = 3x + 5

Expand (multiply out) the brackets, then solve as usual.

EDDIE SAYS
Remember, when expanding brackets, we need to multiply everything in the brackets by the number in front of them. So 4(x - 2) = 4x - 8 Then the rest of the working needs to look something like this: 4x - 8 = 3x + 5 4x - 3x - 8 = 3x - 3x + 5 x - 8 = 5 x - 8 + 8 = 5 + 8 x = 13 Don't let the brackets put you off here - just take it one step at a time.
• Question 9

This equation has brackets on both sides as well as variables on both sides:

5(2 - p) = 2(4 - 2p)

Expand both sets of brackets, then solve as usual.

EDDIE SAYS
So this time we need to expand the brackets on both sides first. 5(2 - p) = 2(4 - 2p) 10 - 5p = 8 - 4p Now, since there are negative p's on both sides, we need to add the smallest number of p's: 10 - 5p = 8 - 4p 10 - 5p + 5p = 8 - 4p + 5p 10 = 8 + p 10 - 8 = 8 - 8 + p 2 = p
• Question 10

Solve the following equation by expanding the brackets first:

3(4y + 3) = 2(3y + 5)

Type your answer below in the form a/b with no spaces, and using the / key to create your fraction bar.

EDDIE SAYS
This is a tricky one! First, expand the brackets: 3(4y + 3) = 2(3y + 5) 12y + 9 = 6y + 10 Then let's subtract the smallest number of y's: 12y - 6y + 9 = 6y - 6y + 10 6y + 9 = 10 6y + 9 - 9 = 10 - 9 6y = 1 6y ÷ 6 = 1 ÷ 6 y = 1/6 Did you remember to leave your answer as a fraction? Well done, you've finished this activity now!
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