Simultaneous equations: multiply both equations

The Maths Habit

1Find the lowest common multiple of 4 and 6.122Multiply every term of 3x − 2y = −5 by −2.−6x + 4y = 1038y = 4, so y = ?y = 124Solve x + 2y = 9 and 3x − 4y = 7.x = 5, y = 2

4 cmTower A6 cmTower B12 cm

Tower A is built from 4 cm blocks. Tower B is built from 6 cm blocks. The two towers are the same height.

How tall could they be? 12 cm, 24 cm, 36 cm, … (multiples of 12)

What is the shortest they could be? How many blocks is that in each? 12 cm: 3 blocks and 2 blocks

To eliminate y, what do you do?

(1)4x+3y=10
(2)5x−2y=1
A(1) × 2 and (2) × 3, then addB(1) × 2 and (2) × 3, then subtractC(1) × 5 and (2) × 4D(1) × 2 only

A: (1) × 2 and (2) × 3, then add

Coefficient
The number in front of a letter.
Lowest common multiple
The smallest number that is a multiple of both numbers.

Solve the simultaneous equations.

(1)3x+2y=4
(2)4x−5y=13
(1) × 5
(3)15x+10y=20
(2) × 2
(4)8x−10y=26
(3) + (4)
23x=46
x=2
(1)6+2y=4
2y=−2
y=−1
x = 2,  y = −1
Check in (2): 8 + 5 = 13 ✓

Cheaper letter: the one that needs fewer multiplications.
The same for both: choose the letter with opposite signs, and add.

Solve the simultaneous equations.

(1)2x+3y=14
(2)3x+2y=11
(1) × 3
(3)6x+9y=42
(2) × 2
(4)6x+4y=22
(3) − (4)
5y=20
y=4
(2)3x+8=11
3x=3
x=1
x = 1,  y = 4
Check in (1): 2 + 12 = 14 ✓

Eliminate x

(1)3x+4y=5
(2)6x−5y=23
(1) × 2
(3)6x+8y=10
(3) − (2)
13y=−13
y=−1
(1)3x−4=5
3x=9
x=3
x = 3,  y = −1
Check in (2): 18 + 5 = 23 ✓

Eliminate y

(1)3x+4y=5
(2)6x−5y=23
(1) × 5
(3)15x+20y=25
(2) × 4
(4)24x−20y=92
(3) + (4)
39x=117
x=3
(1)9+4y=5
4y=−4
y=−1
x = 3,  y = −1
Check in (2): 18 + 5 = 23 ✓

Which letter is cheaper to eliminate? How could you tell before starting? 3 is a factor of 6, so only (1) needs multiplying.

Solve the simultaneous equations.

(1)3x+2y=4
(2)4x−5y=13
(1) × 5
(3)15x+10y=20
(2) × 2
(4)8x−10y=26
(3) + (4)
23x=46
x=2
(1)6+2y=4
2y=−2
y=−1
x = 2,  y = −1
Check in (2): 8 + 5 = 13 ✓

Cheaper letter: the one that needs fewer multiplications.
The same for both: choose the letter with opposite signs, and add.

Solve the simultaneous equations.

(1)5x+2y=1
(2)3x+7y=18
(1) × 7
(3)35x+14y=7
(2) × 2
(4)6x+14y=36
(3) − (4)
29x=−29
x=−1
(1)−5+2y=1
2y=6
y=3
x = −1,  y = 3
Check in (2): −3 + 21 = 18 ✓

Write y = 4 − 3x in the form ax + by = c.

A3x + y = 4B−3x + y = 4C3x − y = 4Dx + y = 4 − 3x

A: 3x + y = 4

Solve the simultaneous equations.

(1)  3y = 2x − 3
(2)5x−2y=24
Rearrange (1)
(3)2x−3y=3
(3) × 2
(4)4x−6y=6
(2) × 3
(5)15x−6y=72
(5) − (4)
11x=66
x=6
(2)30−2y=24
−2y=−6
y=3
x = 6,  y = 3
Check in (1): 3 × 3 = 9
2 × 6 − 3 = 9 ✓

Solve the simultaneous equations.

(1)  4x = 3y − 18
(2)5x+2y=−11
Rearrange (1)
(3)4x−3y=−18
(3) × 2
(4)8x−6y=−36
(2) × 3
(5)15x+6y=−33
(4) + (5)
23x=−69
x=−3
(2)−15+2y=−11
2y=4
y=2
x = −3,  y = 2
Check in (1): 4 × (−3) = −12
3 × 2 − 18 = −12 ✓

Solve the simultaneous equations.

(1)  3y = 2x − 3
(2)5x−2y=24
Rearrange (1)
(3)2x−3y=3
(3) × 2
(4)4x−6y=6
(2) × 3
(5)15x−6y=72
(5) − (4)
11x=66
x=6
(2)30−2y=24
−2y=−6
y=3
x = 6,  y = 3
Check in (1): 3 × 3 = 9
2 × 6 − 3 = 9 ✓

Solve the simultaneous equations.

(1)  4y = 3x − 2
(2)5x+3y=−16
Rearrange (1)
(3)3x−4y=2
(3) × 3
(4)9x−12y=6
(2) × 4
(5)20x+12y=−64
(4) + (5)
29x=−58
x=−2
(2)−10+3y=−16
3y=−6
y=−2
x = −2,  y = −2
Check in (1): 4 × (−2) = −8
3 × (−2) − 2 = −8 ✓

Simultaneous equations: multiply both equations worksheet, page 1

Simultaneous equations: multiply both equations worksheet, page 2

Answer sheet

  1. 3x + 5y = 21 and 5x + 3y = 19. Find the value of x + y without finding x or y.

    5

  2. ax + 3y = 1 and 2x + by = 13 have the solution x = 2, y = −1. Find a and b.

    a = 2, b = −9

  3. Write a pair of simultaneous equations where you must multiply both equations, whose solution is x = 2, y = −3.

    For example 2x + 3y = −5 and 3x + 2y = 0

  1. To eliminate x, multiply (1) by and (2) by , then add or subtract?
    (1)3x+2y=9(2)4x+5y=5

    4 and 3, then subtract

  2. After eliminating y, 6x = 3. Find x.

    x = 12

  3. Solve the simultaneous equations.
    (1)3x+2y=16(2)2x−3y=2

    x = 4, y = 2
Exit tickets to print