Solving Systems by Graphing
Learn how to solve systems of linear equations by graphing both lines and finding their intersection point.
Definition
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Worked Examples
Solve the system by graphing:
Graph the first line
y-intercept: . Slope: (up 1, right 1). Points: , , → Line 1 graphed
Graph the second line
y-intercept: . Slope: (down 1, right 1). Points: , , → Line 2 graphed
Find where the lines intersect
Both lines pass through → Intersection:
Verify the solution
Check : \checkmark Check : \checkmark → Solution verified
Answer: The solution is , meaning and .
Common Mistakes
Reading the intersection point incorrectly from the graph
Why it's wrong: Graphs can be imprecise, especially when the intersection has non-integer coordinates.
Correct: Always verify your answer by substituting back into BOTH original equations.
Forgetting to check the solution in both equations
Why it's wrong: A point might satisfy one equation but not the other.
Correct: The solution must work in BOTH equations. Check each one separately.
Assuming parallel lines have a solution
Why it's wrong: Lines with the same slope but different y-intercepts never intersect.
Correct: If slopes are equal and y-intercepts differ, there is NO solution (inconsistent system).
Graphing errors from incorrect slope calculation
Why it's wrong: Rise over run must be applied correctly. A slope of means down 2, right 1.
Correct: For positive slope: go up and right. For negative slope: go down and right.
Interactive Visual
Linear Function Explorer
| x | y |
|---|---|
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
Interactive Sandbox
Interactive Grapher
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y = 2x + 1
m=2, b=1
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Practice Problems
15 problemsWhat does the solution of a system of equations represent on a graph?
Why It Matters
- Business: Where does revenue equal cost? (break-even point)
- Physics: When do two moving objects meet?
- Economics: Where does supply equal demand? (equilibrium price)
- Planning: When do two schedules or budgets align?
Real World Applications
Break-Even Analysis
Businesses use systems to find when revenue equals costs.
Example:
A company's costs are (500 euros fixed + 2 euros per item) and revenue is (5 euros per item sold). The break-even point is where these lines intersect.
Cost equation: . Revenue equation: .
At what quantity do costs equal revenue?
Step 1: Write the mathematical expression
Set the equations equal:
Meeting Point Problem
Determine when and where two moving objects will meet.
Example:
Car A starts 10 km ahead and travels at 60 km/h. Car B starts at the origin and travels at 80 km/h. Their positions are and . Where do they meet?
Train A: (starts 20 km ahead, 50 km/h). Train B: (starts at origin, 70 km/h).
After how many hours will Train B catch up to Train A?
Step 1: Write the mathematical expression
Set positions equal:
Supply and Demand
Economists find equilibrium where supply equals demand.
Example:
Supply: (price increases as quantity increases). Demand: (price decreases as quantity increases). The equilibrium is where these curves intersect.
Supply: . Demand: .
What is the equilibrium price and quantity?
Step 1: Write the mathematical expression
Set supply equal to demand:
Key Takeaways
- 1To solve a system by graphing, plot both lines and find where they intersect
- 2The intersection point is the solution that satisfies BOTH equations
- 3If lines are parallel (same slope, different y-intercept): NO solution (inconsistent)
- 4If lines are the same: INFINITELY MANY solutions (dependent)
- 5Always verify by substituting the solution into both original equations
Frequently Asked Questions
Glossary
- System of equations
- Two or more equations with the same variables that are solved together
- Solution of a system
- An ordered pair that makes ALL equations in the system true
- Consistent system
- A system that has at least one solution (lines intersect or coincide)
- Inconsistent system
- A system with no solution (parallel lines that never meet)
- Dependent system
- A system where both equations represent the same line (infinitely many solutions)
- Independent system
- A system with exactly one solution (two distinct intersecting lines)