Systems of Linear Equations: Graphing Method

The graphing method for systems

To solve a system graphically: plot both equations on the same axes and find their intersection point. That point is the $(x, y)$ that satisfies both equations simultaneously.

Step-by-step: Solve $y = 2x + 1$ and $y = -x + 7$

Step 1 — Plot Line 1: $y = 2x + 1$. Slope $= 2$, y-intercept $= 1$. Points: $(0, 1)$, $(1, 3)$, $(2, 5)$.

Step 2 — Plot Line 2: $y = -x + 7$. Slope $= -1$, y-intercept $= 7$. Points: $(0, 7)$, $(2, 5)$, $(4, 3)$.

Step 3 — Find intersection. Both lines pass through $(2, 5)$. That's the solution.

Step 4 — Verify algebraically. Set equal: $2x + 1 = -x + 7$ → $3x = 6$ → $x = 2$, $y = 5$ ✓.

Three possible outcomes

• Different slopes → one intersection point (consistent, independent).
• Same slope, different intercepts → parallel → no solution (inconsistent).
• Same slope AND intercept → same line → infinitely many solutions (dependent).

Try it in the visualization

Adjust slopes and intercepts. When lines cross, the intersection is marked. Make slopes equal to see parallel lines (no solution) or identical lines (infinite solutions).