 # Solve Two-Step Equations

In this worksheet, students will learn how to solve two-step equations to find the value of a variable represented by a letter. 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, multiply by 3 then add 7 and the answer is 19. What number am I thinking of?"

You may already be familiar with expressing problems like these algebraically

We can use a letter (usually x) to represent an unknown number, then we can write the problem as an equation.

e.g. 3x + 7 = 19

This equation is a bit more complex, but we go about solving it in the same way as a one-step equation.

We need think of an equation as a set of old fashioned weighing scales which balance: If you add or take something from one side of the scales, it will upset the balance, unless you so the same thing to the other side.

When we solve an equation, we are trying to get the x by itself on the left-hand side.

To do this, we must first remove the + 7 by doing the inverse which is - 7

Then we must remove the 3 in front of the x remembering that this means 3 times x, so the inverse is divide bx y 3

We set this out like this: So the solution to this equation is: x = 4

Let's try another example now.

e.g. 8x - 7 = 9

Remember our scales, what we do to one side of the equation, we must do to the other.

Remember to do the inverse of -7 before the inverse of x8: Remember, x is not always used as the unknown, we could use any letter but the principle is still the same.

Here's one more example before it's over to you.

e.g. 20 - 3w = 8

Now, this is a little trickier as the 3w is being subtracted from the 20, not the other way round.

This is how we need to solve this equation: Remember if you divide a negative by a negative you get a positive.

Now it's over to you.

In this activity, we will solve two-step equations to find the value of a variable represented by a letter.

All the best!

Choose the correct solution to the equation below.

5x + 3 = 28

x = 4

x = 5

x = 6

x = 7

Here's an equation with a different letter:

6g - 1 = 23

Which of the options below is the correct solution?

g = 2

g = 3

g = 4

g = 5

Can you solve the equation below?

11d - 17 = 115

g = 2

g = 3

g = 4

g = 5

Solve the following equation:

9z - 9 = 90

g = 2

g = 3

g = 4

g = 5

Can you match each equation below to their solution?

## Column B

2x + 4 = 18
x = 11
2x - 4 = 18
x = 5
4x - 2 = 18
x = 7
4x - 2 = 18
x = 4

Can you match each equation below to its solution?

## Column B

7q - 15 = -1
q = 2
7q + 15 = 8
q = -1
15q - 7 = 8
q = -2
15q + 7 = -23
q = 1

x = -1 is the solution to at least one of the equations below.

Tick all the equations below which work with this solution.

2x - 1 = 1

1 - 2x = 3

3x - 1 = 2

-2x - 1 = 1

3x - 2 = 1

x = 0.25 is the solution to at least one of the equations below.

Tick all the equations below which work with this solution.

8x - 4 = -2

4x + 9 = 10

6x + 3 = 5.5

20x - 5 = 1

16x + 19 = 21

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

Solve this equation:

-6 - 4x = 6

• Question 1

Choose the correct solution to the equation below.

5x + 3 = 28

x = 5
EDDIE SAYS
Remember that we need to get x on its own first. We have to undo + 3 first and then undo x 5. So our working steps should be: 5x +3 = 28 5x +3 - 3 = 28 - 3 5x = 25 5x ÷5 = 25 ÷5 x = 5 How did you find this first challenge? Review the examples in the Introduction before moving on if you need to.
• Question 2

Here's an equation with a different letter:

6g - 1 = 23

Which of the options below is the correct solution?

g = 4
EDDIE SAYS
The inverse of - 1 is + 1 and the inverse of x 6 is ÷ 6. So we need to follow these steps to find our solution: 6g - 1 = 23 6g - 1 + 1 = 23 + 1 6g = 24 6g ÷ 6 = 24 ÷ 6 g = 4
• Question 3

Can you solve the equation below?

11d - 17 = 115

EDDIE SAYS
This question has some bigger numbers to deal with, but it's still the same method we need to follow. We need to ddd 17 then divide by 11: 11d - 17 = 115 11d - 17 + 17 = 115 + 17 11d = 132 11d ÷11 = 132 ÷11 d = 12 Did you type this number in correctly?
• Question 4

Solve the following equation:

9z - 9 = 90

EDDIE SAYS
Always remember the scales - what we do to one side, we must do to the other! 9z - 9 = 90 9z - 9 + 9 = 90 + 9 9z = 99 9z ÷ 9 = 99 ÷ 9 z = 11 Always show your working so that you can pick up those marks, even if you make a slip in your method.
• Question 5

Can you match each equation below to their solution?

## Column B

2x + 4 = 18
x = 7
2x - 4 = 18
x = 11
4x - 2 = 18
x = 4
4x - 2 = 18
x = 5
EDDIE SAYS
There is a lot to work through here and our working could take up a lot of space, so we'll keep it brief. Here are the inverses in the correct order for each equation: 2x + 4 = 18 → - 4 then ÷ 2 → x = 7 2x - 4 = 18 → + 4 then ÷ 2 → x = 11 4x + 2 = 18 → - 2 then ÷ 4 → x = 4 4x - 2 = 18 → + 2 then ÷ 4 → x = 5 How many matches did you find here?
• Question 6

Can you match each equation below to its solution?

## Column B

7q - 15 = -1
q = 2
7q + 15 = 8
q = -1
15q - 7 = 8
q = 1
15q + 7 = -23
q = -2
EDDIE SAYS
Did you work this out first on paper? Remember in an exam you will be given marks for showing your working. Again, here are our shortened solutions: 7q - 15 = -1 → + 15 then ÷ 7 → q = 2 7q + 15 = 8 → - 15 then ÷ 7 → q = -1 15q - 7 = 8 → + 7 then ÷ 15 → q = 1 15q + 7 = -23 → - 7 then ÷ 15 → q = -2
• Question 7

x = -1 is the solution to at least one of the equations below.

Tick all the equations below which work with this solution.

1 - 2x = 3
-2x - 1 = 1
EDDIE SAYS
We can check each equation by replacing the 'x' in each with '-1' and seeing if it gives the right answer: 2x - 1 = 1 → (2 × -1) - 1 = -3 (wrong) 1 - 2x = 3 → 1 - (2 × -1) = 3 (correct) 3x - 1 = 2 → 3 × -1 - 1 = -4 (wrong) -2x - 1 = 1 → (-2 × -1) - 1 = 1 (correct) 3x - 2 = 1 → 3 × -1 - 2 = -5 (wrong) So two of these options were correct. Did you find them both?
• Question 8

x = 0.25 is the solution to at least one of the equations below.

Tick all the equations below which work with this solution.

8x - 4 = -2
4x + 9 = 10
EDDIE SAYS
Remember, you can check each equation by replacing 'x' with '0.25': 8x - 4 = -2 → 8 × 0.25 -4 = -2 (correct) 4x + 9 = 10 → 4 × 0.25 + 9 = 10 (correct) 6x + 3 = 5.5 → 6 × 0.25 + 3 = 4.4 (wrong) 20x - 5 = 1 → 20 × 0.25 - 5 = 0 (wrong) 16x + 19 = 21 → 16 × 0.25 + 19 = 23 (wrong) So this time only the first two options were correct. How did you do?
• Question 9

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

EDDIE SAYS
Let's work out each equation to see which solution matches: 3a - 3 = -3 → + 3 then ÷ 3 → a = 0 3a + 6 = -3 → - 6 then ÷ 3 → a = -3 10a - 9 = 16 → + 9 then ÷ 10 → a = 2.5 2a - 9 = -8 → + 9 then ÷ 2 → a = 0.5
• Question 10

Solve this equation:

-6 - 4x = 6

EDDIE SAYS
This last one is a tricky one as we need to subtract 4x from -6, not the other way round. We need to start by getting rid of the - 6 by adding 6 to both sides: -6 - 4x = 6 -6 + 6 - 4x = 6 + 6 - 4x = 12 (Don't forget the '-' sign in front of 4x) - 4x ÷ - 4 = 12 ÷ - 4 (We divide both sides by - 4 not just 4) x = -3 (Dividing by - 4 makes the answer negative) Well done, that's a whole exercise completed on solving equations - hopefully you are feeling more confident now? Remember to always write down your workings to ensure you maximise your marks!
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