# Solve Simultaneous Equations (by substitution)

In this worksheet, students will learn how two solve linear simultaneous equations using the substitution method.

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

You may already be familiar with the elimination method to solve two simultaneous equations.

However, in some situations, another method may be preferred.

This is the substitution method, in which one equation is substituted into the other.

This method is what we will focus on in this activity.

If you are not familiar with the elimination method, you may want to review this before trying this activity.

e.g. Solve these equations:

2x + 3y = 30  (1)

y = 3x - 1  (2)

Notice that equation (2) is not written in the same form as equation (1); it is written with y as the subject.

This makes it more suitable for the substitution method.

So, we substitute equation (2) into equation (1).

This means we replace the y in (1) with 3x - 1 as follows:

2x + 3(3x - 1) = 30

Now, this equation only has one variable (x) so it can be solved.

First, expand the brackets:

2x + 9x - 3 = 30

Now, simplify:

11x - 3 = 30

Now, solve:

11x = 33

x = 3

Now, let's substitute x = 3 into either (1) or (2), whichever you think is easiest.

Here, we have substituted into (2):

y = 3 x 3 - 1

y = 8

As before we can check our solutions using equation (1):

2 x 3 + 3 x 8 = 30

6 + 24 = 30 [correct]

So x = 3 and y = 8.

The substitution method is to be preferred over the elimination method when one equation is written in the form y = ... or x = ... and the other is in the form ax + by = c where a, b, c are constants.

Now its time for you to try some!

In this activity, we will solve pairs of simultaneous equation by substituting one equation into the other, then simplifying and solving from there.

We can only do this when one equation is in the form y = ... or x = ... and the other is in the form b.

Make sure you have a pen and paper handy to record your working.

Consider this pair of equations:

a = 3b + 1

7a + 2b = 53

Below you will see the steps to solving these equations, but in the wrong order.

Put the steps into the correct order by matching each to its correct position.

## Column B

1st
a = 7
2nd
23b + 7 = 53 [simplify]
3rd
49 + 4 = 53 [correct]
4th
a = 3 x 2 + 1 [sub into (1)]
5th
a = 3b + 1 (1) and 7a + 2b = 53 (2)
6th
21b + 7 + 2b = 53 [expand]
7th
7(3b + 1) + 2b = 53 [(1) into (2)]
8th
b = 2
9th
23b = 46
10th
7 x 7 + 2 x 2 = 53 [check in (2)]

10p + 11q = 21

p = 5q - 4

Below you will see the steps to solving these equations, but in the wrong order.

Put the steps into the correct order by matching each to its correct position.

## Column B

1st
61q = 61
2nd
p = 5 x 1 - 4 [sub into (2)]
3rd
61q - 40 = 21 [simplify]
4th
10p + 11q = 21 (1) and p = 5q - 4 (2)
5th
10 + 11 = 21 [correct]
6th
10(5q - 4) + 11q = 21 [(2) into (1)]
7th
q = 1
8th
p = 1
9th
50q - 40 + 11q = 21 [expand]
10th
10 x 1 + 11 x 1 = 21 [check in (1)]

Review this new pair of equations:

2x + 3y = 123

x = 3 + 5y ​

Fill in the blanks in the working below to correctly solve these equations using the substitution method

## Column B

1st
61q = 61
2nd
p = 5 x 1 - 4 [sub into (2)]
3rd
61q - 40 = 21 [simplify]
4th
10p + 11q = 21 (1) and p = 5q - 4 (2)
5th
10 + 11 = 21 [correct]
6th
10(5q - 4) + 11q = 21 [(2) into (1)]
7th
q = 1
8th
p = 1
9th
50q - 40 + 11q = 21 [expand]
10th
10 x 1 + 11 x 1 = 21 [check in (1)]

Investigate this next pair of equations:

c = 7d - 2

3c - 4d = 28

Fill in the blanks in the working below to correctly solve these equations using the substitution method

## Column B

1st
61q = 61
2nd
p = 5 x 1 - 4 [sub into (2)]
3rd
61q - 40 = 21 [simplify]
4th
10p + 11q = 21 (1) and p = 5q - 4 (2)
5th
10 + 11 = 21 [correct]
6th
10(5q - 4) + 11q = 21 [(2) into (1)]
7th
q = 1
8th
p = 1
9th
50q - 40 + 11q = 21 [expand]
10th
10 x 1 + 11 x 1 = 21 [check in (1)]

Solve this pair of simultaneous equations using the substitution method:

2w + 7x = 41

w = 3x + 1

Then select the correct solutions from the options given in the table below.

Solve this pair of simultaneous equations using the substitution method:

2a - 9b = 73

b = 3 - 2a

Then select the correct solutions from the options given in the table below.

Solve the following simultaneous equations using the substitution method

3x + 7y = 44

x = 13 - 2y

Then type your solutions in the correct blanks below.

Consider this pair of equations:

3x - y = -10  (1)

5x - y = -16  (2)

To solve these using the method of substitution, we need to convert one equation into the form y = ...

Complete the blanks below to rearrange (1) then solve the equations to find the values of x and y

A father is six times older than his daughter.

Let the father's age be represented byand the daughter's age by d.

a) Create an equation to represent the relationship between f and d.

b) Complete the expression for the father's age in 2 years.

c) Complete the expression for the daughter's age in 2 years.

d) In 2 years time, the father will be 5 times older than his daughter. Complete the second equation connecting f and d.

In the previous question, we found two equations connecting a father's age with his daughter's age:

f = 6d  (1)

f + 2 = 5(d + 2)  (2)

Now, using the method of substitution, solve these equations simultaneously to find the original age of the father and the daughter.

Type your solutions into the blanks below.

• Question 1

Consider this pair of equations:

a = 3b + 1

7a + 2b = 53

Below you will see the steps to solving these equations, but in the wrong order.

Put the steps into the correct order by matching each to its correct position.

## Column B

1st
a = 3b + 1 (1) and 7a + 2b = 53 (...
2nd
7(3b + 1) + 2b = 53 [(1) into (2)...
3rd
21b + 7 + 2b = 53 [expand]
4th
23b + 7 = 53 [simplify]
5th
23b = 46
6th
b = 2
7th
a = 3 x 2 + 1 [sub into (1)]
8th
a = 7
9th
7 x 7 + 2 x 2 = 53 [check in (2)]
10th
49 + 4 = 53 [correct]
EDDIE SAYS
First, we number the equations (1) and (2) and substitute (1) into (2). Next, we expand, simplify and solve the equation to find b. Then, we substitute b = 2 into (1) to find a. Finally, we check by that substituting a = 7 and b= 2 into (2) gives us the correct outcome. How did you find ordering those steps? Review the Introduction now to ensure you have this method locked down, before you move on to the rest of this activity.
• Question 2

10p + 11q = 21

p = 5q - 4

Below you will see the steps to solving these equations, but in the wrong order.

Put the steps into the correct order by matching each to its correct position.

## Column B

1st
10p + 11q = 21 (1) and p = 5q ...
2nd
10(5q - 4) + 11q = 21 [(2) into (...
3rd
50q - 40 + 11q = 21 [expand]
4th
61q - 40 = 21 [simplify]
5th
61q = 61
6th
q = 1
7th
p = 5 x 1 - 4 [sub into (2)]
8th
p = 1
9th
10 x 1 + 11 x 1 = 21 [check in (1...
10th
10 + 11 = 21 [correct]
EDDIE SAYS
First, we number the equations (1) and (2) and substitute (2) into (1). Next, we expand, simplify and solve the equation to find q. Then, we substitute q = 1 into (2) to find p. Finally, we check that substituting q = 1 and p = 1 into (1) gives us the correct outcome.
• Question 3

Review this new pair of equations:

2x + 3y = 123

x = 3 + 5y ​

Fill in the blanks in the working below to correctly solve these equations using the substitution method

EDDIE SAYS
First we number the equations: 2x + 3y = 123 (1) x = 3 + 5y (2) Then we substitute (2) into (1): 2(3 + 5y) + 3y = 123 Expand the brackets: 6 + 10y + 3y = 123 Simplify: 6 + 13y = 123 And solve: 13y = 117 y = 9 Now let's substitute into (2) to find x: x = 3 + 5 x 9 x = 48 Finally, let's check that both these values work in (1): 2 x 48 + 3 x 9 = 123 96 + 27 = 123 [correct] How are you doing? Are you getting the hang of this method now?
• Question 4

Investigate this next pair of equations:

c = 7d - 2

3c - 4d = 28

Fill in the blanks in the working below to correctly solve these equations using the substitution method

EDDIE SAYS
First we number the equations: c = 7d - 2 (1) 3c - 4d = 28 (2) And we substitute (1) into (2): 3(7d -2) - 4d = 28 Expand our brackets: 21d - 6 - 4d = 28 Simplify: 17d - 6 = 28 Solve: 17d = 34 d = 2 Substitute into this value of d into (1) to find c: c = 7 x 2 - 2 c = 12 Check that these values give the correct outcome in (2): 3 x 12 - 4 x 2 = 28 36 - 8 = 28 (correct)
• Question 5

Solve this pair of simultaneous equations using the substitution method:

2w + 7x = 41

w = 3x + 1

Then select the correct solutions from the options given in the table below.

EDDIE SAYS
First number the equations: 2w + 7x = 41 (1) w = 3x + 1 (2) 2(3x + 1) + 7x = 41 [sub (2) into (1)] 6x + 2 + 7x = 41 [expand] 13x + 2 = 41 [simplify] 13x = 39 [solve] x = 3 w = 3 × 3 + 1 [sub into (2)] w = 10 2 × 10 + 7 × 3 = 41 [check in (1)] 20 + 21 = 41 [correct] How did you get on tackling this one more independently?
• Question 6

Solve this pair of simultaneous equations using the substitution method:

2a - 9b = 73

b = 3 - 2a

Then select the correct solutions from the options given in the table below.

EDDIE SAYS
First let's number the equations: 2a - 9b = 73 (1) b = 3 - 2a (2) 2a - 9(3 - 2a) = 73 [sub (2) into (1)] 2a - 27 + 18a = 73 [expand] 20a - 27 = 73 [simplify] 20a = 100 [solve] a = 5 b = 3 - 2 × 5 [sub into (2)] b = -7 2 × 5 - 9 × -7 = 73 [check in (1)] 10 + 63 = 73 [correct]
• Question 7

Solve the following simultaneous equations using the substitution method

3x + 7y = 44

x = 13 - 2y

Then type your solutions in the correct blanks below.

EDDIE SAYS
Compare your working with ours below: 3x + 7y = 44 (1) x = 13 - 2y (2) 3(13 - 2y) + 7y = 44 [sub (2) into (1)] 39 - 6y + 7y = 44 39 + y = 44 y = 5 x = 13 - 2 × 5 [sub into (2)] x = 3 3 × 3 + 7 × 5 = 44 [check in (1)] 9 + 35 = 44 [correct]
• Question 8

Consider this pair of equations:

3x - y = -10  (1)

5x - y = -16  (2)

To solve these using the method of substitution, we need to convert one equation into the form y = ...

Complete the blanks below to rearrange (1) then solve the equations to find the values of x and y

EDDIE SAYS
Here are our starting, numbered equations: 3x - y = -10 (1) 5x - y = -16 (2) To rearrange (1), add y to both sides and add 10 to both sides: 3x + 10 = y (3) Now substitute (3) into (2): 5x - (3x + 10) = -16 Expand our brackets: 5x - 3x - 10 = -16 2x - 10 = -16 2x = -6 x = -3 Now let's substitute x into (3) to find y: y = 3 × -3 + 10 y = 1 Did you use your pen and paper to record your working accurately? Don't worry if you made a slip in this complicated method. Examiners recognise that there is a lot to remember here, and award marks for your working, as well as getting the correct answer, to be fair.
• Question 9

A father is six times older than his daughter.

Let the father's age be represented byand the daughter's age by d.

a) Create an equation to represent the relationship between f and d.

b) Complete the expression for the father's age in 2 years.

c) Complete the expression for the daughter's age in 2 years.

d) In 2 years time, the father will be 5 times older than his daughter. Complete the second equation connecting f and d.

EDDIE SAYS
a) The father is 6 times older than his daughter so: f = 6d b) The father will be 2 years older, so we need to add 2 to f to show this: f + 2 c) The daughter will be 2 years older so we need to add 2 to d to show this: d + 2 d) It's two years in the future for the father (f + 2), and he is 5 times older than his daughter's age then (d + 2), so: f + 2 = 5 (d + 2) OK so we haven't solved any equations here, but wait until question 10 to bring this all together!
• Question 10

In the previous question, we found two equations connecting a father's age with his daughter's age:

f = 6d  (1)

f + 2 = 5(d + 2)  (2)

Now, using the method of substitution, solve these equations simultaneously to find the original age of the father and the daughter.

Type your solutions into the blanks below.

EDDIE SAYS
Compare your working with ours written here: f = 6d (1) f + 2 = 5(d + 2) (2) Sub (1) into (2): 6d + 2 = 5(d + 2) Expand: 6d + 2 = 5d + 10 Subtract 5d from both sides: d + 2 = 10 d = 8 Sub d into (1) to find f: f = 6 x 8 f = 48 The father was 48 and the daughter was 8 originally. That's it, your last question is complete! Simultaneous equations are a tricky concept, so you may want to revise the Level 1 or 2 activities in this group if you would like some more practise.
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