Solving simultaneous equations graphically

Two linear equations in \(x\) and \(y\) considered together are called simultaneous linear equations. Their solution is the pair of values of \(x\) and \(y\) that makes both equations true at the same time. On a graph, that pair is the point where the two lines intersect.

Why: every point on the line \(x+4y=6\) satisfies the first equation, and every point on the line \(5x-y=9\) satisfies the second. Only a point on both lines satisfies both equations, and the only such point is the intersection \((2,1)\).

Method

1. For each equation, make a table of three points.
2. Choose scales so that both lines fit on the grid and the graph fills most of it. The two axes may have different scales; avoid awkward scales such as \(1\) cm for \(3\) units.
3. Draw both lines on the same axes and label each line with its equation.
4. Read both coordinates of the point of intersection. The solution is \(x=\) the \(x\)-coordinate and \(y=\) the \(y\)-coordinate.
5. Check by substituting the pair into both equations.

Example

Solve \(x+4y=6\) and \(5x-y=9\) graphically.

The lines meet at \((2,1)\), so \(x=2\) and \(y=1\). Check: \(2+4(1)=6\) and \(5(2)-1=9\).

Check: a graphical answer is exact only when the lines cross at a grid point. Otherwise read each coordinate as accurately as the scale allows (often to \(1\) decimal place) and say that it is an estimate. Always give both coordinates.