Systems of Equations: Graphing Method

The graphing method

The graphical method finds the solution to a system of equations by plotting both equations on the same coordinate plane and finding where they intersect. The intersection point is the $(x, y)$ that satisfies both equations simultaneously.

Step-by-step solution

We have:

• Line 1: $y = -x + 5$ (slope $= -1$, y-intercept $= 5$)
• Line 2: $y = 2x - 1$ (slope $= 2$, y-intercept $= -1$)

Step 1 — Plot Line 1. Start at $(0, 5)$ on the y-axis. Slope is $-1$, so go right 1, down 1 to $(1, 4)$, then $(2, 3)$, $(3, 2)$, etc.

Step 2 — Plot Line 2. Start at $(0, -1)$ on the y-axis. Slope is $2$, so go right 1, up 2 to $(1, 1)$, then $(2, 3)$, $(3, 5)$, etc.

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

Step 4 — Verify algebraically. Set the two expressions equal: $-x + 5 = 2x - 1$. Add $x$ to both sides: $5 = 3x - 1$. Add 1: $6 = 3x$. Divide: $x = 2$. Then $y = 2(2) - 1 = 3$.

Step 5 — Check. Line 1: $y = -(2) + 5 = 3$ ✓. Line 2: $y = 2(2) - 1 = 3$ ✓.

Solution: $(x, y) = (2, 3)$.

Three possible outcomes

• One intersection point (different slopes): exactly one solution — the system is consistent and independent.
• No intersection (parallel lines, same slope, different intercepts): no solution — the system is inconsistent.
• Infinite intersections (same line): infinitely many solutions — the system is dependent.

Limitations of the graphing method

The graphing method gives an approximate answer when the intersection has non-integer coordinates. For exact answers, use substitution or elimination. But graphing provides valuable geometric intuition about what the algebraic solution means.

Try it in the visualization

Adjust the slope and intercept of each line. When the lines cross, the intersection point appears automatically. Make the slopes equal to see parallel lines (no solution). Make both lines identical to see infinite solutions.