A-Level MathsSimultaneous Equations


Example –

Solve the simultaneous equations: x^2+y^2=10 and x+2y=5.

This example requires solution via substitution, i.e. make either x or y the subject of one equation and insert it into the other. The obvious choice would be to make x the subject of the second equation – it is the quickest, least complicated choice. The second equation tells us that x=5-2y. We can insert this into the first equation: (5-2y)^2+y^2=10. By multiplying out the brackets and simplifying we see that this is a quadratic equation in y:

(5-2y)^2+y^2=10
\Longrightarrow (5-2y)(5-2y)+y^2=10
\Longrightarrow 25-10y-10y+4y^2+y^2=10
\Longrightarrow 5y^2-20y+15=0
\Longrightarrow y^2-4y+3=0
\Longrightarrow (y-3)(y-1)=0
This tells us that y has to be either 3 or 1. If y=3, then x=5-2\times 3=-1 (from the second equation) and if y=1 then x=5-2\times 1=3.

We obtain the solutions (x_1,y_1)=(-1,3) and (x_2,y_2)=(3,1).