# Equations: Solutions to Harder Equations with Two Unknowns

In this worksheet, students must find pairs of numbers that satisfy a harder equation with two unknowns.

Key stage:  KS 2

Curriculum topic:   Algebra

Curriculum subtopic:   Use Equations

Difficulty level:

### QUESTION 1 of 10

In this worksheet, we look at equations with two unknowns.

a and b stand for two numbers, which could be different or the same.

Look at this equation.

a + 2b = 8

It tells you that a added to double b is 8.

If a is 6, then b will be 1 because 6 + 2 × 1 = 6 + 2 = 8.

If a = 2, then b = 3 because 2 + 2 × 3 = 2 + 6 = 8.

If a = 9, then b must be negative. b will be because 9 + 2 × -½ = 9 + -1 = 8.

If a = 1, then b = 3½ because 1 + 2 × 3½ = 1 + 7 = 8.

Example

3a - 2b = 7

Which of these solutions satisfy the equation?

a = 7, b = 0

a = 7, b = 7

a = 4, b = 2½

a = 5, b = 3

Put the given pairs into the equation to see if 3a - 2b is 7 or not.

In each case, we work out 3 × a take away 2 × b to see if the answer is 7 or not.

 a = 7, b = 0 does not fit because 3 × 7 - 2 × 0 = 21 - 0 = 21 (not 7) a = 7, b = 7 fits because 3 × 7 - 2 × 7 = 21 - 14 = 7 a = 4, b = 2½ fits because 3 × 4 - 2 × 2½ = 12 - 5 = 7 a = 5, b = 3 does not fit because 3 × 5 - 2 × 3 = 15 - 6 = 9 (not 7)

a + 2b = 7

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 1, b = -3

a = 5, b = 1

a = 3, b = 2

a = 4, b = 3

2a - b = 5

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 8, b = 11

a = 3, b = 1

a = 5, b = 10

a = 3½, b = 2

b - 2a = 6

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 3, b = 12

a = 2, b = 10

a = -1, b = 4

a = ½, b = 7

4a + b = 12

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 3, b = -2

a = 3, b = 2

a = 2, b = -2

a = 0, b = 12

6a - b = 12

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 2½, b = 3

a = 0, b = 12

a = 2, b = 0

a = 1, b = 6

3a + 2b = 15

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 5, b = 0

a = 3, b = 3

a = 2, b = 3

a = 4, b = 1½

5a - 2b = 0

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 10, b = -2

a = 2, b = 5

a = 0, b = 0

a = 4, b = 10

4a - 3b = 6

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 6, b = 5

a = 3, b = 4

a = 9, b = 10

a = 10½, b = 8

3a - 5b = 2

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 9, b = 5

a = 5, b = 10

a = 4, b = 2

a = 4, b = -2

7a + 2b = -5

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = -3, b = 8

a = -5, b = 15

a = 3, b = -13

a = -8, b = 25

• Question 1

a + 2b = 7

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 5, b = 1
a = 3, b = 2
EDDIE SAYS
1 + 2 × -3 = 1 + -6 = -5
5 + 2 × 1 = 5 + 2 = 7
3 + 2 × 2 = 3 + 4 = 7
4 + 2 × 3 = 4 + 6 = 10
• Question 2

2a - b = 5

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 8, b = 11
a = 3, b = 1
a = 3½, b = 2
EDDIE SAYS
2 × 8 - 11 = 16 - 11 = 5
2 × 3 - 1 = 6 - 1 = 5
2 × 5 - 10 = 10 - 10 = 0
2 × 3½ - 2 = 7 - 2 = 5
• Question 3

b - 2a = 6

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 3, b = 12
a = 2, b = 10
a = -1, b = 4
a = ½, b = 7
EDDIE SAYS
12 - 2 × 3 = 12 - 6 = 6
10 - 2 × 2 = 10 - 4 = 6
4 - 2 × -1 = 4 - -2 = 6
7 - 2 × ½ = 7 - 1 = 6
• Question 4

4a + b = 12

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 0, b = 12
EDDIE SAYS
4 × 3 + -2 = 12 - 2 = 10
4 × 3 + 2 = 12 + 2 = 14
4 × 2 + -2 = 8 - 2 = 6
4 × 0 + 12 = 0 + 12 = 12
• Question 5

6a - b = 12

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 2½, b = 3
a = 2, b = 0
EDDIE SAYS
6 × 2½ - 3 = 15 - 3 = 12
6 × 0 - 12 = 0 - 12 = -12
6 × 2 - 0 = 12 - 0 = 12
6 × 1 - 6 = 6 - 6 = 0
• Question 6

3a + 2b = 15

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 5, b = 0
a = 3, b = 3
a = 4, b = 1½
EDDIE SAYS
3 × 5 + 2 × 0 = 15 + 0 = 15
3 × 3 + 2 × 3 = 9 + 6 = 15
3 × 2 + 2 × 3 = 6 + 6 = 12
3 × 4 + 2 × 1½ = 12 + 3 = 15
• Question 7

5a - 2b = 0

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 2, b = 5
a = 0, b = 0
a = 4, b = 10
EDDIE SAYS
5 × 10 - 2 × -2 = 50 - -4 = 54
5 × 2 - 2 × 5 = 10 - 10 = 0
5 × 0 - 2 × 0 = 0 - 0 = 0
5 × 4 - 2 × 10 = 20 - 20 = 0
• Question 8

4a - 3b = 6

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 9, b = 10
EDDIE SAYS
4 × 6 - 3 × 5 = 24 - 15 = 9
4 × 3 - 3 × 4 = 12 - 12 = 0
4 × 9 - 3 × 10 = 36 - 30 = 6
4 × 10½ - 3 × 8 = 42 - 24 = 18
• Question 9

3a - 5b = 2

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = 9, b = 5
a = 4, b = 2
EDDIE SAYS
3 × 9 - 5 × 5 = 27 - 25 = 2
3 × 5 - 5 × 10 = 15 - 50 = -35
3 × 4 - 5 × 2 = 12 - 10 = 2
3 × 4 - 5 × -2 = 12 - -10 = 22
• Question 10

7a + 2b = -5

Which of these solutions satisfy the equation?

Tick all the solutions that fit.

a = -3, b = 8
a = -5, b = 15
a = 3, b = -13
EDDIE SAYS
7 × -3 + 2 × 8 = -21 + 16 = -5
7 × -5 + 2 × 15 = -35 + 30 = -5
7 × 3 + 2 × -13 = 21 + -26 = -5
7 × -8 + 2 × 25 = -56 + 50 = -6
---- OR ----

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