Simultaneous equations: forming and solving

The Maths Habit

1The cost in pence of 3 pens at p pence each and 2 rulers at r pence each3p + 2r2Two numbers add to 20 and differ by 6. Find them.13 and 73Solve 2x + 3y = 22 and y = x − 1x = 5, y = 44The graphs of two equations cross at (−1, 4). What is the solution of the pair?x = −1, y = 4

A torch and a pack of batteries cost £9 together. What could each one cost?

Lots of answers: £5 and £4, £8 and £1, £4.50 and £4.50, …

The torch costs £5 more than the batteries. Now what could each one cost?

Only one answer: the torch costs £7 and the batteries £2.

A club has 5 times as many juniors (j) as coaches (c). Which equation is right?

A j = 5cB 5j = cC j + c = 5D j = c + 5

A. Test it: 2 coaches means 10 juniors, and 5 × 2 = 10 ✓. B gives 5j = 2: fewer juniors than coaches.

Complete each equation.

13 pencils at p pence and 2 rubbers at r pence cost 95p.  3p + 2r = 9524 stamps at s pence and 1 envelope at e pence cost £3.10.   + e = 3104s32 bags of rice at r grams and 1 bag of oats at t grams weigh 2.5 kg.  2r + t = 25004There are 3 times as many forks (f) as spoons (s).  f = 3s5A mug costs 60p more than a coaster.  m c = 60−  (m − c = 60)612 tickets are sold: a adult tickets and c child tickets.   = 12a + c

Bulbs all cost the same. You want to find the price of one bulb. Which is the best first line?

A Let b = the cost of one bulb in penceB b = bulbsC b = the number of bulbs

A: a letter stands for a number, here a cost, with its unit.

A pen and a ruler cost 90p. Two pens and two rulers cost £1.80. Can you find each price?

A YesB No

B: the second fact is the first one doubled. It is the same fact, so any pair that adds to 90p fits both.

4 seed packets and 3 bulbs cost £10.10. 2 seed packets and 5 bulbs cost £10.30. Find the cost of each.

1
Let lineLets= the cost of one seed packet in penceb= the cost of one bulb in pence
2
Table
seed packetsbulbspence
(1)4s+3b=1010
(2)2s+5b=1030
3
Two different facts?Yes: (2) is not (1) multiplied by a number.
4
TestTry s = 10, b = 1: 4 packets and 3 bulbs would cost 43p, and 4 × 10 + 3 × 1 = 43 ✓
5
Solve
(2) × 2
(3)4s+10b=2060
(3) − (1)
7b=1050
b=150
(1)4s+450=1010
4s=560
s=140
s = 140,  b = 150
Check in (2): 280 + 750 = 1030 ✓
6
Sentence: does it make sense?A seed packet costs £1.40 and a bulb costs £1.50. Both sensible prices ✓

3 bags of compost and 2 bags of sand weigh 64 kg. 1 bag of compost and 4 bags of sand weigh 48 kg. Find the weight of each bag.

1
Let lineLetc= the weight of one bag of compost in kgs= the weight of one bag of sand in kg
2
Table
compostsandkg
(1)3c+2s=64
(2)c+4s=48
3
Two different facts?Yes: (2) is not (1) multiplied by a number.
4
TestTry c = 10, s = 1: 3 bags at 10 kg and 2 at 1 kg weigh 32 kg, and 3 × 10 + 2 × 1 = 32 ✓
5
Solve
(1) × 2
(3)6c+4s=128
(3) − (2)
5c=80
c=16
(2)16+4s=48
4s=32
s=8
c = 16,  s = 8
Check in (1): 48 + 16 = 64 ✓
6
Sentence: does it make sense?A bag of compost weighs 16 kg and a bag of sand weighs 8 kg. Sensible ✓

A jar holds 20 coins. They are all 5p and 10p coins, and they are worth £1.40. How many of each coin are there?

1
Let lineLetf= the number of 5p coinst= the number of 10p coins
2
Table
5p coins10p coinstotal
(1)f+t=20
(2)5f+10t=140
3
Two different facts?Yes: (1) counts coins, (2) counts pence.
4
TestTry f = 15, t = 5: 20 coins worth 75p + 50p = 125p, and 15 + 5 = 20 ✓, 5 × 15 + 10 × 5 = 125 ✓
5
Solve
(1) × 5
(3)5f+5t=100
(2) − (3)
5t=40
t=8
(1)f+8=20
f=12
f = 12,  t = 8
Check in (2): 60 + 80 = 140 ✓
6
Sentence: does it make sense?There are twelve 5p coins and eight 10p coins. Whole numbers of coins ✓

Simultaneous equations: forming and solving worksheet, page 1

Answer sheet

  1. An equilateral triangle has sides 2x + y, x + 3y and 15 cm. Find x and y.

    x = 6, y = 3

  2. 20 kg of coffee is a blend of beans at £12 a kg and beans at £18 a kg. The blend costs £14.70 a kg. How much of each is in it?

    11 kg at £12 and 9 kg at £18

  3. A quiz has 20 questions. A right answer scores 3 points and a wrong answer loses 1 point. Ali answers every question and scores 36. How many does Ali get right?

    14 right (and 6 wrong)

  1. There are 3 times as many red counters (r) as blue (b). Write an equation.

    r = 3b

  2. 2 kites and 3 frisbees cost £19. 1 kite and 1 frisbee cost £7.50. Let k and f be the costs in pounds. Write two equations, find f, then use (1) to find k. Answer in a sentence.

    A kite costs £3.50 and a frisbee £4.

  3. A scarf is 30 g heavier than a hat. 3 scarves and 4 hats weigh 1140 g. Let s and h be the weights in grams. Find each.

    A scarf weighs 180 g and a hat 150 g.
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