Lesson 7.2
Solving by Graphing
If a solution is where two lines meet, the easiest way to find it is to draw a picture.
Introduction
Graphing is the most visual way to solve a system. We graph the first line, graph the second line, and point to the spot where they crash into each other.
Past Knowledge
Lesson 5.8 (Graphing Slope-Intercept). You must be able to graph quickly.
Today's Goal
Find the coordinate of the intersection.
Future Success
This visual understanding helps when we do "Substitution" and "Elimination" later. You'll know why we are doing the algebra.
Key Concepts
The Three Outcomes
The lines cross at exactly one point. Most common.
Parallel lines. They never touch, so there is no answer.
They are the same line! Every point works.
Worked Examples
Example 1: The Perfect Cross
BasicSolve the system by graphing.
Step 1: Graph
Line 1 starts at 2, goes up 1 over 1.
Line 2 starts at 4, goes down 1 over 1.
Step 2: Inspect
They cross exactly at .
Solution:
Example 2: Parallel Lines
IntermediateSolve the system.
Observation
Notice the slopes. Both are .
The y-intercepts are different ($1$ and $-3$).
Conclusion: No Solution.
They are parallel railroad tracks. They will NEVER cross.
Example 3: Standard Form Trick
AdvancedSolve the system.
Step 1: Graphing Standard Form
For : Intercepts are and .
For : Intercept is . Use slope .
Solution:
The lines meet at .
Common Pitfalls
Messy Intersections
Graphing is terrible if the answer is . It's hard to see decimals. That's why we will learn algebraic methods (Substitution/Elimination) next.
Drawing Sloppy Lines
If your lines are wiggly or you don't use a ruler, you will miss the intersection point. Be precise!
Real-Life Applications
Break-Even Point:
- Business Owners graph "Cost" and "Revenue" lines.
- The point where they cross is the Break-Even Point.
- Below that point, you lose money. Above it, you make profit.
Practice Quiz
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