Solving linear simultaneous equations

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

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)

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

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

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)

4𝑥+3𝑦=11 2𝑥−5𝑦=25 $0

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

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.