1. Solving linear simultaneous equations

2. Elimination method Solve the simultaneous equations 2𝑎+3𝑏=16 5𝑎−3𝑏=−2
𝒂=𝟐
𝒃=𝟒
(1)
(2)
Add the two equations together:
7𝑎=14
𝑎=2
Substitute 𝑎=2 into (1):
2×2 +3𝑏=16
3𝑏+4=16
3𝑏=12
𝑏=4

3. Elimination method Solve the simultaneous equations 3𝑝+2𝑞=7 4𝑝+3𝑞=9
𝒑=𝟑
𝒒=−𝟏
(1)
(2)
Subtract (4) from (3):
𝑝=3
Multiply (1) by 3:
9𝑝+6𝑞=21
(3)
Substitute 𝑝=3 into (1):
3×3 +2𝑞=7
2𝑞=−2
𝑞=−1
Multiply (2) by 2:
8𝑝+6𝑞=18
(4)

4. Substitution method Solve the simultaneous equations 5𝑒+6𝑓=8 𝑒+4𝑓=3
𝒆=𝟏
𝒇=𝟎.𝟓
(1)
(2)
15−20𝑓+6𝑓=8
−14𝑓=−7
𝑓=0.5
Rearrange (2):
𝑒=3−4𝑓
(3)
Substitute 𝑓=0.5 into (3):
𝑒=3− 4×0.5
𝑒=3−2
𝑒=1
Substitute 𝑒=3−4𝑓 into (1):
5 3−4𝑓 +6𝑓=8

5. Problems in context (1)
Simon and Jess go grocery shopping. Jess buys 3 apples and 6 bananas for £1.95. Simon buys 5 apples and 1 banana for £1.90. Determine the cost of one apple and one banana.
Students can use either elimination or substitution to solve.
Solution: apple: 35p, banana: 15p

6. Problems in context (2)
Find the coordinates of the point where the straight lines
4𝑥+3𝑦=11
and
2𝑥−5𝑦=25
intersect.
Students are most likely to use elimination to solve.
Solution: (5, −3)

7. 4𝑥+3𝑦=11
2𝑥−5𝑦=25 $0

8. Extension Solve the simultaneous equations 𝑎+𝑏+𝑐=0 𝑎−𝑏+𝑐=−4 𝑎+𝑏−2𝑐=9
Solution: 𝑎=1, 𝑏=2, 𝑐=−3

9. Spot the mistake Solve the simultaneous equations 2𝑥+3𝑦=−16 3𝑥−2𝑦=−11
(1)
(2)
Multiply (1) by 2: 4𝑥+6𝑦=−32
Multiply (2)by 3: 9𝑥−6𝑦=−33
Subtract (3) from (4): 5𝑥=−1
𝑥=−0.2
Substitute 𝑥=−0.2 into (1):
−0.4+3𝑦=−16
3𝑦=−15.6
𝑦=−5.2
(3)
(4)
Signs for 6𝑦 are different so equations should be added, not subtracted to eliminate the y terms
Correct solution is 𝑥=−5, 𝑦=−2.