Systems of Equations: Graphing Method
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn how to solve a system of two linear equations by graphing both lines on the same coordinate plane and identifying their intersection point as the solution.
The Concept
Section titled “The Concept”The graphing method solves a system by plotting both equations as lines on the same graph. The solution is the point (x, y) where the lines intersect, satisfying both equations at once.
Steps:
- Graph the first equation (use slope-intercept, intercepts, or table method)
- Graph the second equation on the same axes
- Find the intersection point (read x and y coordinates)
- Write the solution as (x, y)
- Check by substituting into both original equations
Types of solutions:
- One intersection → one solution
- Lines parallel (same slope, different y-intercept) → no solution
- Lines coincide (same equation) → infinite solutions
Graphing is visual and great for checking, but less precise for fractional answers.
Worked Example
Section titled “Worked Example”Solve the system by graphing:
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First line: y = 2x − 3 y-intercept −3 at (0, −3), slope 2 → point (1, −1)
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Second line: y = −x + 6 y-intercept 6 at (0, 6), slope −1 → point (1, 5)
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Intersection: set 2x − 3 = −x + 6 → 3x = 9 → x = 3, y = 2(3) − 3 = 3
Solution: (3, 3)
Check:
- First: 2(3) − 3 = 6 − 3 = 3 ✓
- Second: −3 + 6 = 3 ✓
Both true → solution (3, 3).
Real-World Application
Section titled “Real-World Application”Graphing systems helps compare options visually:
- Two phone plans: Plan A: y = 40 + 0.10x (per text), Plan B: y = 30 + 0.15x. Graph both to find when costs are equal
- Two savings accounts: A: y = 200 + 50m, B: y = 100 + 70m (m = months). Intersection finds when balances match
- Supply and demand: price where quantity supplied = quantity demanded
The graph shows trends (which plan is cheaper before/after intersection) and the exact meeting point.