Solving Systems of Equations by Graphing
A system of equations asks for the point that satisfies both equations at once. Graphed, that is exactly where the two lines cross.
Graphing makes the solution visible and also shows the special cases: parallel lines never cross (no solution), and identical lines overlap everywhere (infinitely many).
The intersection is the solution
Graph both lines on the same axes. The coordinates of their crossing point satisfy both equations, so that point is the solution.
Check it by substituting back into both equations — a correct solution makes both true. Graphing is quickest when the intersection lands on clean integer coordinates.
The two lines below cross at a single point ; because that point lies on both lines, its coordinates are the one solution that satisfies both equations.
How many solutions?
Two lines that cross once give exactly one solution. Parallel lines (same slope, different intercept) never meet, so there is no solution.
If the two equations describe the same line, they overlap at every point — infinitely many solutions. Comparing slopes and intercepts predicts which case you are in.
Worked examples
Example 1: reading the intersection
Lines and cross where?
Answer:
Example 2: no solution
How many solutions does , have?
Answer: No solution
Example 3: infinitely many solutions
How many solutions does , have?
Answer: Infinitely many solutions
Try one yourself
Common questions
What does the intersection point represent?
The one (x, y) pair that satisfies both equations simultaneously — the system's solution.
When is there no solution?
When the lines are parallel: same slope, different -intercepts. They never cross, so no point works for both.
What does 'infinitely many solutions' mean?
The two equations are really the same line. Every point on it satisfies both, so there are infinitely many solutions.
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