Simultaneous equations: what a solution is

The Maths Habit

  1. Work out 3x + 2y when x = 4 and y = −1.

    10

  2. Work out 5a − 2b when a = 2 and b = −3.

    16

  3. Is x = 6 a solution of 4x − 7 = 17?

    Yes

  4. Solve 2y + 9 = 3.

    y = −3

  5. Find the point on y = 5 − x where x = 2.

    (2, 3)

Receipt 1

1 muffin

1 smoothie

Total£4.20

Receipt 2

1 muffin

2 smoothies

Total£6.70

What does a smoothie cost?

What is different between the receipts?

What does the difference cost?

One more smoothie costs £6.70 − £4.20 = £2.50

A smoothie costs £2.50 and a muffin costs £1.70

Let m = the cost of one muffin in pence

Let s = the cost of one smoothie in pence

m + s = 420m + 2s = 670

m = 170, s = 250

Check in the second: 170 + 2 × 250 = 670 ✓

Simultaneous equations
Two or more equations that are true at the same time.
Solution
The values of the letters that make every equation true.

Complete the table of pairs.

x + y = 10

x1472.512
y9637.5−2

Complete the table. Find the pair that is in both tables.

x − y = 4

x45678
y01234

Pair in both: x = 7, y = 3

Complete the table of pairs.

x + y = 10

x1472.512
y9637.5−2

Complete both tables. Find the pair that is in both.

x + y = 8

x01234
y87654

y − x = 2

x01234
y23456

Pair in both: x = 3, y = 5

012345678910012345678910xyx + y = 10x − y = 4

Plot the pairs from both tables.

x + y = 10x − y = 4

The pair in both, (7, 3), is where the lines cross.

Which pair solves both equations?

x + 2y = 113x − y = 5

A(5, 3)B(2, 1)C(3, 4)D(4, 3)
C (3, 4)

Does (3, 2) solve both equations?

(1)  2x + y = 8(2)  x − 4y = −5

(1)2 × 3 + 2 = 8 ✓

(2)3 − 4 × 2 = −5 ✓

Solves both? Yes

Does (4, −1) solve both equations?

(1)  x + 3y = 1(2)  2x − y = 7

(1)4 + 3 × (−1) = 1 ✓

(2)2 × 4 − (−1) = 9 ≠ 7 ✗

Solves both? No

Does (3, 2) solve both equations?

(1)  2x + y = 8(2)  x − 4y = −5

(1)2 × 3 + 2 = 8 ✓

(2)3 − 4 × 2 = −5 ✓

Solves both? Yes

Does (−2, 5) solve both equations?

(1)  3x + y = −1(2)  y − x = 7

(1)3 × (−2) + 5 = −1 ✓

(2)5 − (−2) = 7 ✓

Solves both? Yes

Simultaneous equations: what a solution is worksheet, page 1

Simultaneous equations: what a solution is worksheet, page 2

Answer sheet

Write a pair that fits x + y = 12

Check your pair in x − y = 2

Stand up if it fits both.

(7, 5): 7 + 5 = 12 ✓ and 7 − 5 = 2 ✓

  1. The pair (4, 1) solves both x + ay = 11 and 2x + y = 9. Find a.

    a = 7

  2. Write a pair of equations whose solution is x = 5 and y = −2. Check them.

    e.g. x + y = 3 and x − y = 7

  3. 1 tea and 1 cake cost £4. 1 tea and 3 cakes cost £8.40. Find the cost of a cake.

    £2.20 (a tea is £1.80)

  1. Does (5, −2) solve both 2x + 3y = 4 and x − y = 7?

    Yes

  2. Does (1, 6) solve both 4x + y = 10 and 2x + y = 9?

    No: 2 × 1 + 6 = 8 ≠ 9

  3. The pair (k, 3) fits x + 2y = 10. Find k. Does it also fit 3x − y = 9?

    k = 4, yes

  4. Write down a pair that fits x + y = 9 but not y = 2x. Use tables to find the pair that fits both.

    e.g. (5, 4); both: x = 3, y = 6
Exit tickets to print