Solving systems by graphing

Every linear equation draws a straight line. A system of two equations draws two lines, and the solution is the point where they cross — because that point lies on both lines at once.

Reading the solution

Draw both lines on the same grid. The coordinates of the crossing point are the values of x and y that solve the system. If the lines cross at (2, 3), then x = 2 and y = 3.

Special cases

• If the two lines are parallel, they never cross — the system has no solution.
• If the two equations draw the same line, every point fits — there are infinitely many solutions.

When graphing is best

Graphing is great for seeing what a solution means and for whole-number answers. For exact non-whole answers, substitution or elimination is more reliable.

Three rules that always help

• The solution is the point where the two lines cross.
• Parallel lines → no solution; the same line twice → infinitely many.
• Read both coordinates of the crossing point off the axes.

Keep going

• Practice: Solve a system from a graph
• Back to the systems overview