# Solve Two-Step Equations (including brackets)

In this worksheet, students will learn how to solve two-step equations by finding inverse operations, plus performing them in the correct order, to reach accurate answers.

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

"I think of a number, subtract 6 then multiply by 7 and the answer is 63. What number am I thinking of?"

You may already be familiar with this sort of puzzle, and the fact that we can express problems like this algebraically

However, although the example above is very similar to equations you may have learnt about before, there is an important difference in the order of operations.

In this puzzle, the subtraction needs to be performed before the multiplication, and this makes a difference when we write it as an equation.

Normally, multiplication and division calculations are performed before addition and subtraction.

If we want to change this order, we need to use brackets.

So in order to write this puzzle as an equation with x as the unknown number we would write:

e.g. 7(x - 6) = 63

The x - 6 needs to be inside brackets, as we want to calculate this part first.

The 7 outside the bracket means 'times 7' and this is the second step.

This type of equation can still be solved using the balancing method, where we always do the same to both sides to get the x by itself.

Since the x 7 was performed last, we need to 'undo' this first by ÷ 7, then we undo the - 6 by + 6.

We should set out our working like this:

So, if we return to the opening statement, the number we were thinking of is 15

Here is another equation, which is a little more complicated.

e.g. 3(2x + 9) = 51

First, the x is multiplied by 2 (2x) then 9 is added (2x + 9) and finally, it is multiplied by 3.

So we need to use the inverses of these processes, in the reverse order.

Here is what our working needs to look like:

Now, technically this is a three-step equation, but it uses the same method as the two-step ones; there's just an extra line of working.

Are you ready to try solving some yourself now?

In this activity, we will solve two- or three-step equations where we need to find the inverse operations, plus perform them in the correct order, to reach accurate answers.

Make sure you have a pen and paper handy so that you can record your working as it becomes more complex.

Don't be tempted to use a calculator though; rely on your mental maths power!

Choose the correct solution to the equation below:

2(x + 8) = 20

x = 2

x = 6

x = 14

x = 16

Here's an equation with a different letter:

3(h - 1) = 21

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

h = 6

h = 7

h = 8

h = 9

Can you solve the equation below?

5(t - 9) = 70

h = 6

h = 7

h = 8

h = 9

Solve the following equation:

7(p + 2) = 84

h = 6

h = 7

h = 8

h = 9

Can you match each of the equations below to their solutions?

## Column B

2(3x - 4) = -8
x = 3
3(2x - 4) = -6
x = 1
4(3x - 2) = 16
x = 2
4(2x - 3) = 12
x = 0

Can you match each equation below to its solution?

## Column B

3(3y - 7) = 24
y = 4
5(2y + 4) = 0
y = -2
5(4y + 3) = 95
y = 1
9(2y + 1) = 27
y = 5

x = 19 is the solution to at least one of the equations below.

Tick all the options which fit this solution.

4(2x - 27) = 48

7(x - 11) = 56

5(3x + 1) = 290

9(4x - 69) = 54

x = ½ is the solution to at least one of the equations below.

Tick all the options which fit this solution.

2(x + 4) = 11

4(x - 3) = -10

3(4x - 1) = 3

3(2x + 1) = 9

2(9x - 2) = 5

Can you match the equations below to their correct solutions?

Solve the equation below:

2(400 - 3x) = 326

• Question 1

Choose the correct solution to the equation below:

2(x + 8) = 20

x = 2
EDDIE SAYS
We can describe this equation as: "Add 8 to x then multiply this answer by 2." 8 has been added first, then we have multiplied by 2, so the inverse of this is: ÷ 2 then - 8. Remember to set out your working correctly and clearly: 2(x + 8) = 20 2(x + 8) ÷ 2 = 20 ÷ 2 x + 8 = 10 x + 8 - 8 = 10 - 8 x = 2 How did you find that one? Take your time to review the Introduction now if you need to before moving on to the rest of this activity.
• Question 2

Here's an equation with a different letter:

3(h - 1) = 21

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

h = 8
EDDIE SAYS
How did you do? Remember the letter is not important, it just stands for something we don't know. The inverse of '× 3' is '÷ 3' and the inverse of - 1 is + 1. So we can find our solution using these steps: 3(h - 1) = 21 3(h - 1) ÷ 3 = 21 ÷ 3 h - 1 = 7 h - 1 + 1 = 7 + 1 h = 8
• Question 3

Can you solve the equation below?

5(t - 9) = 70

EDDIE SAYS
To solve this one, we need to ÷ 5 then + 9. 5(t - 9) = 70 5(t - 9) ÷ 5 = 70 ÷ 5 t - 9 = 14 t - 9 + 9 = 14 + 9 t = 23 Did you type this number in correctly?
• Question 4

Solve the following equation:

7(p + 2) = 84

EDDIE SAYS
Don't forget to start by getting rid of the number outside the bracket. 7(p + 2) = 84 7(p + 2) ÷ 7 = 84 ÷ 7 p + 2 = 12 p + 2 - 2 = 12 - 2 p = 10
• Question 5

Can you match each of the equations below to their solutions?

## Column B

2(3x - 4) = -8
x = 0
3(2x - 4) = -6
x = 1
4(3x - 2) = 16
x = 2
4(2x - 3) = 12
x = 3
EDDIE SAYS
Did you spot that these are all three-step equations? There is a lot to work through here, so our working can take up a lot of space, so we'll keep it as brief as possible here. Here are the inverses in the correct order for each equation: 2(3x - 4) = -8 → ÷ 2 then + 4 then ÷ 3 → x = 0 3(2x - 4) = -6 → ÷ 3 then +4 then ÷ 2 → x = 1 4(3x - 2) = 16 → ÷ 4 then + 2 then ÷ 3 → x = 2 4(2x - 3) = 12 → ÷ 4 then + 3 then ÷ 2 → x = 3 Did you find all of these matches?
• Question 6

Can you match each equation below to its solution?

## Column B

3(3y - 7) = 24
y = 5
5(2y + 4) = 0
y = -2
5(4y + 3) = 95
y = 4
9(2y + 1) = 27
y = 1
EDDIE SAYS
Some more 3-step equations to work through here. Again, here are shortened solutions: 3(3y - 7) = 24 → ÷ 3 then + 7 then ÷ 3 → y = 5 5(2y + 4) = 0 → ÷ 5 then - 4 then ÷ 2 → y = -2 5(4y + 3) = 95 → ÷ 5 then - 3 then ÷ 4 → y = 4 9(2y + 1) = 27 → ÷ 9 then - 1 then ÷ 2 → y = 1 How are you doing with these? Don't worry if you making some mistakes, so long as you can see why you got it wrong. Also, remember that your working can earn you marks, even if you don't always arrive at the correct answer.
• Question 7

x = 19 is the solution to at least one of the equations below.

Tick all the options which fit this solution.

7(x - 11) = 56
5(3x + 1) = 290
EDDIE SAYS
There are some big numbers here, but don't be tempted to reach for a calculator! We can check each equation by replacing 'x' with '19' in each case, and seeing if it gives the right answer: 4(2x - 27) = 44 → 4x(2 × 19 - 27) = 4 × 11 = 44 (wrong) 7(x - 11) = 56 → 7x(19 - 11) = 7 × 8 = 56 (correct) 5(3x + 1) = 195 → 5(3 × 19 + 1) = 5 × 58 = 290 (correct) 9(4x - 69) = 54 → 9x(4 × 19 - 69) = 9 × 7 = 63 (wrong) So two of these equations worked with this solution. Did you find them both?
• Question 8

x = ½ is the solution to at least one of the equations below.

Tick all the options which fit this solution.

4(x - 3) = -10
3(4x - 1) = 3
2(9x - 2) = 5
EDDIE SAYS
Remember, you can check each equation by replacing x with ½: 2(x + 4) = 11 → 2 × (½ + 4) = 9 (wrong) 4(x - 3) = -10 → 4 × (½ - 3) = -10 (correct) 3(4x - 1) = 3 → 3 × (4x½ - 1) = 3 × 1 = 3 (correct) 3(2x + 1) = 9 → 3 × (2x½ + 1) = 3 × 2 = 6 (wrong) 2(9x - 2) = 5 → 2 × (9x½ - 2) = 2 × 2.5 = 5 (correct) So this time there were three equations which matched this solution. How did you do?
• Question 9

Can you match the equations below to their correct solutions?

EDDIE SAYS
Let's work out each equation to see which solution matches: 3(3n - 4) = 33 → ÷ 3 then + 4 then ÷3 → n = 5 2(2n + 3) = 6 → ÷ 2 then - 3 then ÷ 2 → n = 0 5(2n + 1) = 20 → ÷ 5 then - 1 then ÷ 2 → n = 1.5 42 = 3(4 + 5n) → ÷ 3 then - 4 then ÷ 5 → n = 2 Don't be put off that this last one is written 'back to front'. It's just the same as the others really.
• Question 10

Solve the equation below:

2(400 - 3x) = 326

EDDIE SAYS
This is a very challenging one! There are some big numbers here, but our method is still the same: 2 × (400 - 3x) = 326 2(400 - 3x) ÷ 2 = 326 ÷ 2 400 - 3x = 163 400 - 3x - 400 = 163 - 400 - 3x = - 237 (Don't forget the '-' in front of the 237 here) - 3x ÷ -3 = - 237 ÷ - 3 (We have to divide by - 3 not just 3) x = 79 Well done, if you got this one right. And that's a whole exercise finished - excellent!
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