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Solve Simultaneous Equations (by substitution)

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

'Solve Simultaneous Equations (by substitution)' worksheet

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:  

Worksheet Overview

QUESTION 1 of 10

In previous activities, we have used 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. The following example will demonstrate this method.

Solve the 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, substitute x = 3 into either (1) or (2) whichever you think is easiest. Here, we substitute 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.

 

 

Below you will see all the steps to solving the equations

a = 3b + 1

and 7a + 2b = 53

but in the wrong order. Put the steps into the correct order.

 

Column A

Column B

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

Below you will see all the steps to solving the equations

10p + 11q = 21

and  p = 5q - 4

but in the wrong order. Put the steps into the correct order.

 

Column A

Column B

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

Using the substitution method, solve the simultaneous equations

2x + 3y = 123

and 

x = 3 + 5y 

by filling in the blanks below.

Column A

Column B

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

Using the substitution method, solve the simultaneous equations

c = 7d - 2

and 

3c - 4d = 28

by filling in the blanks below.

Column A

Column B

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

For the pair of simultaneous equations below, select the correct solutions from the options given.

2w + 7x = 41

w = 3x + 1

 

For the pair of simultaneous equations below, select the correct solutions from the options given.

2a - 9b = 73

b = 3 - 2a

 

On a piece of paper solve the following simultaneous equations using the substitution method. Fill in your solutions in the blanks.

3x + 7y = 44

and

x = 13 - 2y

 

 

 

The simultaneous equations 

3x - y = -10  (1)

5x - y = -16  (2)

are to be solved using the method of substitution. To do this equation (1) must be written in the form y = ......

In the blank below complete the rearranged equation 2.

Now solve the equations on paper and fill in the solutions below.

A father is six times older than his daughter. Let the father's age be 'f' and the daughter's age be 'd'. 

a) Complete the equation using 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 is 5 times older than the daughter. Complete the second equation connecting f and d.

In question 9 we found two equations connecting a father's age with his daughter's age. They were

f = 6d  (1)

and 

f + 2 = 5(d + 2)  (2)

Now, using the method of substitution solve the equations simultaneously to find the original age of the father and the daughter. Do your working on paper and fill in the solutions below.

  • Question 1

Below you will see all the steps to solving the equations

a = 3b + 1

and 7a + 2b = 53

but in the wrong order. Put the steps into the correct order.

 

CORRECT ANSWER

Column A

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 sub b = 2 into (1) to find a. Finally, we check by substituting a = 7 and b= 2 into (2).
  • Question 2

Below you will see all the steps to solving the equations

10p + 11q = 21

and  p = 5q - 4

but in the wrong order. Put the steps into the correct order.

 

CORRECT ANSWER

Column A

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 by substituting q = 1 and p = 1 into (1).
  • Question 3

Using the substitution method, solve the simultaneous equations

2x + 3y = 123

and 

x = 3 + 5y 

by filling in the blanks below.

CORRECT ANSWER
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 substitute into (2) x = 3 + 5 x 9 x = 48 Finally, check in (1) 2 x 48 + 3 x 9 = 123 96 + 27 = 123 [correct] How are you getting on?
  • Question 4

Using the substitution method, solve the simultaneous equations

c = 7d - 2

and 

3c - 4d = 28

by filling in the blanks below.

CORRECT ANSWER
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 21d - 6 - 4d = 28 Simplify 17d - 6 = 28 Solve 17d = 34 d = 2 Substitute into (1) c = 7 x 2 - 2 c = 12 Check in (2) 3 x 12 - 4 x 2 = 28 36 - 8 = 28 (correct)
  • Question 5

For the pair of simultaneous equations below, select the correct solutions from the options given.

2w + 7x = 41

w = 3x + 1

 

CORRECT ANSWER
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 x 3 + 1 [sub into (2)] w = 10 2 x 10 + 7 x 3 = 41 [check in (1)] 20 + 21 = 41 [correct]
  • Question 6

For the pair of simultaneous equations below, select the correct solutions from the options given.

2a - 9b = 73

b = 3 - 2a

 

CORRECT ANSWER
EDDIE SAYS
First 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 x 5 [sub into (2)] b = -7 2 x 5 - 9 x -7 = 73 [check in (1)] 10 + 63 = 73 [correct]
  • Question 7

On a piece of paper solve the following simultaneous equations using the substitution method. Fill in your solutions in the blanks.

3x + 7y = 44

and

x = 13 - 2y

 

 

 

CORRECT ANSWER
EDDIE SAYS
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 x 5 [sub into (2)] x = 3 3 x 3 + 7 x 5 = 44 [check in (1)] 9 + 35 = 44 [correct]
  • Question 8

The simultaneous equations 

3x - y = -10  (1)

5x - y = -16  (2)

are to be solved using the method of substitution. To do this equation (1) must be written in the form y = ......

In the blank below complete the rearranged equation 2.

Now solve the equations on paper and fill in the solutions below.

CORRECT ANSWER
EDDIE SAYS
3x - y = -10 (1) 5x - y = -16 (2) To rearrang (1) add y to both sides and add 10 to both sides 3x + 10 = y (1) Now sub (1) into (2) 5x - (3x + 10) = -16 5x - 3x - 10 = -16 [expand] 2x - 10 = -16 2x = -6 x = -3 y = 3 x -3 + 10 y = 1
  • Question 9

A father is six times older than his daughter. Let the father's age be 'f' and the daughter's age be 'd'. 

a) Complete the equation using 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 is 5 times older than the daughter. Complete the second equation connecting f and d.

CORRECT ANSWER
EDDIE SAYS
a) The father is 6 times older than the daughter so f = 6d b) The father will be 2 years older so add 2 to f makes f + 2 c) The daughter will be 2 years older so add 2 to d makes d + 2 d) The father is f+2 and he is 5 times older than the daughter who is d+2 so f+2 = 5(d+2) OK so you haven't solved any equations here, but wait till question 10!
  • Question 10

In question 9 we found two equations connecting a father's age with his daughter's age. They were

f = 6d  (1)

and 

f + 2 = 5(d + 2)  (2)

Now, using the method of substitution solve the equations simultaneously to find the original age of the father and the daughter. Do your working on paper and fill in the solutions below.

CORRECT ANSWER
EDDIE SAYS
f = 6d (1) f + 2 = 5(d + 2) (2) We can 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 into (1) f = 6 x 8 f = 48 The father was 48 and the daughter was 8 originally. That\'s it. All done
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