Home > Equations > Simultaneous Equations: Substitution Method

## Simultaneous Equations: Substitution Method

This is exactly the same method we use to find where two lines intersect.

Process:

1. Rearrange one of the equations to make one of the variables a subject
2. Substitute for this variable into the other equation
3. Solve the resulting equation
4. Substitute these solutions into one of the original equations to find the values of the other variable

Example

Solve the system of equations

$y=4x-3\rightarrow 1$

$3y+x=7\rightarrow 2$

Substitute eqn 1 into eqn 2

$\implies 3(4x-3)+x=7$

$\displaystyle \implies 12x-9+x=7\implies 13x=16\implies \frac{16}{13}$

Substituting into eqn 1 gives

$\displaystyle y=4\left(\frac{16}{13}\right)-3=\frac{25}{13}$

Solve the system of equations

$y-3=x^2+4x\rightarrow 1$

$2x=y-6\rightarrow 2$

Rearrange eqn 2 to make y the subject

$\implies y=2x+6\rightarrow 3$

Substitute eqn 3 into eqn 1

$\implies (2x+6)-3=x^2+4x\implies x^2+2x-3=0$

Factorising gives

$(x+3)(x-1)=0\implies x=-3\mbox{ or }x=1$

Substituting these values into eqn 3 gives

$y=0\mbox{ and }y=8$

So the solutions are $(-3,0)\mbox{ and }(1,8)$

Solve the system of equations

$3x+2y=13\rightarrow 1$

$2x-y=-3\rightarrow 2$

Rearranging eqn 2 to make y the subject gives

$y = 2x+3\rightarrow 3$

Substituting eqn 3 into eqn 1 gives

$3x+2(2x+3)=13\implies 7x+6=13\implies x=1$

Substituting the value of x into eqn 3 gives

$y=5$