Solving a System by Graphing

In this lesson you will: A solution of a system is an ordered pair that satisfies every equation in it; Graph both lines; the point where they cross is the solution; Check a candidate pair in both equations before you write it down.

Two conditions, one point

A single linear equation has infinitely many solutions — every point on its line. Put two equations side by side and ask for the pairs that satisfy both, and the field of candidates collapses. Two lines that are not parallel meet exactly once.

So the picture is simple: draw both lines, find the crossing point, read off its coordinates. The answer to a system is a pair, (x,y), not a single number.

Get both equations into y = form first

Graphing a line written as y=mx+b takes about five seconds: plot the intercept, count the slope, draw. Graphing 2x+y=8 as written takes longer and invites mistakes.

Solve each equation for y before you touch the graph paper. 2x+y=8 becomes y=-2x+8; x-y=1 becomes y=x-1. Now both are ready.

Reading a crossing point honestly

Graphing is the method that shows you what is going on, and the method that lies to you most easily. If the crossing point sits at (2,3), a hand-drawn graph finds it. If it sits at (2.2,4.6), a hand-drawn graph gets you close and no closer.

Use the picture to understand the system and to estimate. When the problem wants an exact answer and the lines do not cross on a lattice point, switch to the algebra in Lessons 12.2 and 12.3.

Common mistake — Checking the pair in only one equation

A student decides that (2,4) solves the system y=x+2 and y=3x-1, because 2+2=4 checks out.

It is a reasonable-looking piece of work. The substitution was done correctly, the arithmetic is right, and one equation really is satisfied.

But (2,4) only lies on the first line. In the second, 3(2)-1=5, not 4. A point that satisfies one equation is a point on one line, and there are infinitely many of those. A solution of the system has to be on both lines at once.

How to catch it: Substitute into both equations, every time, and write both checks down. If either one fails, the pair is not a solution — and a pair that fails one check almost always means you read the crossing point off the graph a square too far.