Solving by drawing

To draw a line you only need two points. The easiest two: put x = 0 and find y, then put y = 0 and find x.

For x + 3y = 6: x = 0 gives y = 2, and y = 0 gives x = 6. So the line goes through (0, 2) and (6, 0).

For 2x – 3y = 12: x = 0 gives y = –4, and x = 3 gives y = –2. So (0, –4) and (3, –2).

Draw both lines on the same graph. The point where they cross is the solution. Here they meet at (6, 0), so x = 6, y = 0. Put them back into both equations to check.

Champa's sale. Skirts are two less than twice the pants: y = 2x – 2. Skirts are also four less than four times the pants: y = 4x – 4. The lines cross at (1, 0): she bought 1 pair of pants and no skirt. A strange answer, but it fits both sentences.

What the other two cases look like:

• 5x – 8y + 1 = 0 and 3x – (24/5)y + 3/5 = 0: multiply the second by 5/3 and you get the first again. It is one line drawn twice, so infinitely many solutions.
• Parallel lines never cross: no solution.

Two special lines: x = 3 is a straight line parallel to the y-axis, and y = –4 is parallel to the x-axis. They meet at (3, –4). And x = 0 is the y-axis itself.

When the graph is a poor tool: if the answer is something like (–1.75, 3.3), it is easy to misread the graph. That is why we also learn the two algebra methods next.