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

Write the final answer

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.

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