# Solving Simultaneous Equations by Substitution (2)

In this worksheet, students solve simultaneous equations by substituting one equation into the other.

Key stage:  KS 4

Curriculum topic:  Algebra

Curriculum subtopic:  Solve Simultaneous Equations in 2 Variables

Difficulty level:

### QUESTION 1 of 10

Simultaneous equations involve two variables, each of whose value must be found.

The following method, in which one equation is substituted into the other, works best when one equation contains a single letter on its own.

Example

Solve simultaneously

3x - 2y = 14

y - 2x = -9

3x - 2y = 14     (a)

y - 2x = -9        (b)

We find the equation with the single letter, i.e. (b), which has a single y in it.

We rearrange this to read y = ...

y = 2x - 9     (b)

Substitute (b) into (a).

Wherever we see a y in (a) we will replace it with (2x - 9), remembering the brackets.

(a) becomes:

3x - 2(2x - 9) = 14

Multiply out the brackets and solve.

Be careful with the double negative signs.

Notice that we now have an equation just in x.

3x - 4x + 18 = 14

-x + 18 = 14

-x = -4

x = 4

Now use (b) to determine y.

y = 2x - 9

When x = 4, this gives:

y = 2×4 - 9 = 8 - 9 = -1

y = -1

Check by putting this into the other equation (a) to get 3 × 4 - 2 × -1 = 12 + 2 = 14

Solution is x = 4, y = -1

Solve simultaneously:

3x + 7y = 0

x - y = 10

x = 3, y = 7

x = 7, y = -3

x = 21, y = 1

Solve simultaneously:

5x + 3y = 27

y + 2x = 10

x = 3, y = 4

x = -6, y = -1

x = 1, y = 3

Solve simultaneously:

6x + 5y = 56

y - 2x = 8

x = 1, y = 10

x = 1½, y = 4

x = -1, y = 7

Solve simultaneously:

3x + 2y = 13

y = 4x - 10

x = 4, y = 1

x = 12, y = 3

x = 3, y = 2

Solve simultaneously:

y = 3x - 2

13x - 5y = 6

x = 4, y = 2

x = 2, y = 4

x = 6, y = 16

Solve simultaneously:

4x - 3y = 9

y + 3x = 10

x = 2, y = 1

x = 3, y = 1

x = 4, y = 2

Solve simultaneously:

2x + 5y = 31

3y + x = 18

x = 3, y = 5

x = 13, y = 1

x = 4, y = 6

Solve simultaneously:

7x - 2y = 64

3x + y = 7

x = 2, y = 3

x = 6, y = -11

x = 5, y = 5

Solve simultaneously:

5x + 6y = 17

y - 5x = -3

x = 2, y = 4

x = 3, y = 4

x = 1, y = 2

Solve simultaneously:

4x - 3y = 10

2y + x = -14

x = 4, y = 2

x = 7, y =6

x = -2, y = -6

• Question 1

Solve simultaneously:

3x + 7y = 0

x - y = 10

x = 7, y = -3
• Question 2

Solve simultaneously:

5x + 3y = 27

y + 2x = 10

x = 3, y = 4
• Question 3

Solve simultaneously:

6x + 5y = 56

y - 2x = 8

x = 1, y = 10
• Question 4

Solve simultaneously:

3x + 2y = 13

y = 4x - 10

x = 3, y = 2
• Question 5

Solve simultaneously:

y = 3x - 2

13x - 5y = 6

x = 2, y = 4
• Question 6

Solve simultaneously:

4x - 3y = 9

y + 3x = 10

x = 3, y = 1
• Question 7

Solve simultaneously:

2x + 5y = 31

3y + x = 18

x = 3, y = 5
• Question 8

Solve simultaneously:

7x - 2y = 64

3x + y = 7

x = 6, y = -11
• Question 9

Solve simultaneously:

5x + 6y = 17

y - 5x = -3

x = 1, y = 2
• Question 10

Solve simultaneously:

4x - 3y = 10

2y + x = -14