Solving by Graphing
Solve a linear system by finding the ordered pair shared by both graphs, then verify that the visual intersection really satisfies both equations.
How can the graph solve both equations at once?
Each graph shows every solution to one equation. Their intersection automatically keeps only the ordered pairs that belong to both solution sets.
The method changes the representation, not the definition of a system solution.
The crossing point is a candidate. Both equations verify it.
For y = x + 1 and y = −x + 5, the graphs cross at (2, 3). Reading the graph gives the candidate; substitution explains why it is valid.
Graph both equations
Every point on each line satisfies its own equation.
Read the intersection
The lines share the point (2, 3).
Verify both constraints
3 = 2 + 1 and 3 = −2 + 5, so the point survives both equations.
Locate the point shared by both lines.
A graph can reveal structure without always revealing exact coordinates.
The system y = x and y = −x + 1 intersects at (1/2, 1/2). On a coarse grid that location may only be estimated. Substitution and elimination can recover exact coordinates when the graph is visually ambiguous.