Simultaneous equations: add or subtract

The Maths Habit

1Solve 3d + 5 = 26d = 72Simplify 2y − (−y)3y3Simplify −3y − (−3y)04Is x = 5, y = −1 a solution of both x + y = 4 and 3x − y = 16?Yes: 5 + (−1) = 4 and 15 + 1 = 16

Work out the price of a doughnut.£2

Work out the price of a milkshake.£3

Write how you worked this out using equations.

3d + 2m = 122d + 2m = 10

Work out the price of a doughnut.£3

Work out the price of a milkshake.£2.50

Write how you worked this out using equations.

3d + 4m = 192d + 4m = 16

Work out the price of a milkshake.£2.50

Work out the price of a doughnut.£1.50

Write how you worked this out using equations.

4d + 3m = 13.54d + 2m = 11

Work out the price of a milkshake.£1.50

Work out the price of a doughnut.£2.50

Write how you worked this out using equations.

2d + 3m = 9.52d + m = 6.5

is now dis now m

dddddmm= £13
dddmm= £9

Work out d.d = 2

Work out m.m = 1.5

Write each row as an equation.

5d + 2m = 133d + 2m = 9

Work out the price of a doughnut. £2

Work out the price of a milkshake. £3

d = 200, m = 300

+= £12
+= £10
dddmm= £12
ddmm= £10

Let d = the cost of one doughnut in pence, m = the cost of one milkshake in pence.

3d + 2m = 1200
2d + 2m = 1000

3d + 2m = £14d + 2m = £8

Find d and m.

d = £3, m = £2.50
2d + 5m = £14.502d + 3m = £9.50

Find d and m.

d = £1, m = £2.50
4d + m = 17d + m = 8

Find d and m.

d = 3, m = 5

2y = 1, so y = ?

A 2B 12C −1D 1

B: 12

Coefficient
The number in front of a letter.
Eliminate
Remove a letter, so one equation with one letter is left.

Solve the simultaneous equations.

(1)4x+y=19
(2)x+y=7
(1) − (2)
3x=12
x=4
(2)4+y=7
y=3
x = 4,  y = 3
Check in (1): 16 + 3 = 19 ✓

Solve the simultaneous equations.

(1)6x+y=32
(2)2x+y=12
(1) − (2)
4x=20
x=5
(2)10+y=12
y=2
x = 5,  y = 2
Check in (1): 30 + 2 = 32 ✓

Solve the simultaneous equations.

(1)4x+y=19
(2)x+y=7
(1) − (2)
3x=12
x=4
(2)4+y=7
y=3
x = 4,  y = 3
Check in (1): 16 + 3 = 19 ✓

Solve the simultaneous equations.

(1)5x−3y=12
(2)2x−3y=3
(1) − (2)
3x=9
x=3
(2)6−3y=3
−3y=−3
y=1
x = 3,  y = 1
Check in (1): 15 − 3 = 12 ✓

To eliminate y, do you add, subtract or multiply first?

(1)5x+2y=1
(2)3x−2y=7
A AddB SubtractC Multiply first

A: add

Solve the simultaneous equations.

(1)4x+y=19
(2)x−y=1
(1) − (2)
3x+2y=18nothing removed
(1) + (2)
5x=20
x=4
(1)16+y=19
y=3
x = 4,  y = 3
Check in (2): 4 − 3 = 1 ✓

The same term, sign included: subtract.
Opposite terms (y and −y, 3y and −3y): add.

Solve the simultaneous equations.

(1)4x+3y=18
(2)2x−3y=0
(1) + (2)
6x=18
x=3
(2)6−3y=0
−3y=−6
y=2
x = 3,  y = 2
Check in (1): 12 + 6 = 18 ✓

Solve the simultaneous equations.

(1)4x+y=19
(2)x−y=1
(1) − (2)
3x+2y=18nothing removed
(1) + (2)
5x=20
x=4
(1)16+y=19
y=3
x = 4,  y = 3
Check in (2): 4 − 3 = 1 ✓

The same term, sign included: subtract.
Opposite terms (y and −y, 3y and −3y): add.

Solve the simultaneous equations.

(1)2x+y=10
(2)−2x+3y=14
(1) + (2)
4y=24
y=6
(1)2x+6=10
2x=4
x=2
x = 2,  y = 6
Check in (2): −4 + 18 = 14 ✓

Add to eliminate y

(1)4x+2y=21
(2)4x−2y=19
(1) + (2)
8x=40
x=5
(2)20−2y=19
−2y=−1
y=12
x = 5,  y = 12
Check in (1): 20 + 1 = 21 ✓

Subtract to eliminate x

(1)4x+2y=21
(2)4x−2y=19
(1) − (2)
4y=2
y=12
(2)4x−1=19
4x=20
x=5
x = 5,  y = 12
Check in (1): 20 + 1 = 21 ✓

Which letter would you remove first, and does it matter?

Simultaneous equations: add or subtract worksheet, page 1

Answer sheet

  1. 2x + 3y = 17 and 3x + 2y = 18. Find the value of x + y without finding x or y.

    7

  2. x + y = 10,  y + z = 13 and x + z = 11. Find x, y and z.

    x = 4, y = 6, z = 7

  3. Solve 4x + 3y = 2 and 4x − 3y = 14 twice: once by adding, once by subtracting. Do you get the same answer?

    Yes: x = 2, y = −2

  1. Add or subtract to eliminate y? Decide only, do not solve.(a)(1)7x−2y=3(2)4x−2y=6(b)(1)3x+4y=11(2)x−4y=−7

    (a) subtract  (b) add

  2. Solve the simultaneous equations.
    (1)6x+2y=7(2)2x+2y=3

    x = 1, y = 12

  3. Solve the simultaneous equations.
    (1)3x+2y=4(2)5x−2y=12

    x = 2, y = −1
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