Simultaneous Equations

Algebra and Sequences

From a described situation

Introduce a letter for each unknown and state precisely what each represents, including units
Each separate fact in the question becomes one equation
Solve, then answer in context rather than leaving the letters bare

Exam tip

Multiply every term when scaling an equation, the constant included, and show the scaled pair before eliminating — that step carries a method mark. Decide between adding and subtracting from the signs of the matched coefficients. Substitute your answers into the equation you did not use: it catches an error in one line.

Worked example

Forming and solving from a context

Two coffees and three teas cost £11.40. Four coffees and one tea cost £13.80. Find the cost of each drink.

Solution:

Let c be the cost of a coffee and t the cost of a tea, both in pounds
The two facts give 2c + 3t = 11.40 and 4c + t = 13.80
Rearrange the second: t = 13.80 − 4c
Substitute into the first: 2c + 3(13.80 − 4c) = 11.40
Expand: 2c + 41.40 − 12c = 11.40, so −10c = −30 and c = 3.00
Then t = 13.80 − 12.00 = 1.80
A coffee costs £3.00 and a tea costs £1.80
Check: 2(3.00) + 3(1.80) = 11.40