Mathematics » Equations and Inequalities » Simultaneous Equations Continued

Solving by Substitution II

Solving by Substitution

• Use the simplest of the two given equations to express one of the variables in terms of the other.

• Substitute into the second equation. By doing this we reduce the number of equations and the number of variables by one.

• We now have one equation with one unknown variable which can be solved.

• Use the solution to substitute back into the first equation to find the value of the other unknown variable.

Example

Question

Solve for $$x$$ and $$y$$: \begin{align*} y-2x &= -4 \qquad \ldots (1) \\ x^2 + y &= 4 \qquad \ldots (2) \end{align*}

Make $$y$$ the subject of the first equation

$y = 2x – 4$

Substitute into the second equation and simplify

\begin{align*} x^2 + (2x-4) &= 4 \\ x^2 + 2x – 8 &= 0 \end{align*}

Factorise the equation

\begin{align*} (x+4)(x-2) &= 0 \\ \therefore x = -4 &\text{ or } x = 2 \end{align*}

Substitute the values of $$x$$ back into the first equation to determine the corresponding $$y$$-values

If $$x = -4$$: \begin{align*} y &= 2(-4)-4 \\ &= -12 \end{align*}

If $$x = 2$$: \begin{align*} y &= 2(2) – 4 \\ &= 0 \end{align*}

Check that the two points satisfy both original equations

The solution is $$x = -4 \text{ and } y = -12$$ or $$x = 2 \text{ and } y = 0$$. These are the coordinate pairs for the points of intersection as shown below.