Solving Systems by Graphing

Learn how to solve systems of linear equations by graphing both lines and finding their intersection point.

Intermediate25 minLesson

Definition

A system of linear equations is a set of two or more linear equations with the same variables. To solve a system by graphing:

1.Graph each equation on the same coordinate plane
2.Find the point where the lines intersect
3.The intersection point $(x, y)$ is the solution

\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}

The solution satisfies both equations simultaneously.

Try it now

If two lines are parallel, how many solutions does the system have?

Worked Examples

Solve the system by graphing:

\begin{cases} y = x + 1 \\ y = -x + 5 \end{cases}

Graph the first line $y = x + 1$

y-intercept: $(0, 1)$ . Slope: $1$ (up 1, right 1). Points: $(0, 1)$ , $(1, 2)$ , $(2, 3)$ → Line 1 graphed

Graph the second line $y = -x + 5$

y-intercept: $(0, 5)$ . Slope: $-1$ (down 1, right 1). Points: $(0, 5)$ , $(1, 4)$ , $(2, 3)$ → Line 2 graphed

Find where the lines intersect

Both lines pass through $(2, 3)$ → Intersection: $(2, 3)$

Verify the solution

Check $y = x + 1$ : $3 = 2 + 1$ \checkmark Check $y = -x + 5$ : $3 = -2 + 5$ \checkmark → Solution verified

Answer: The solution is $(2, 3)$ , meaning $x = 2$ and $y = 3$ .

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 $-2$ 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

y = x

Slope (m)1

Y-Intercept (b)0

run

rise

x	y
-2	-2
-1	-1
0	0
1	1
2	2

Interactive Sandbox

Interactive Grapher

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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History

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Practice Problems

15 problems

Problem 1 of 15

Easy

What does the solution of a system of equations represent on a graph?

Why It Matters

Graphing systems of equations helps you visualize solutions and understand relationships between equations:

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?

Seeing the intersection makes abstract algebra concrete!

Real World Applications

Break-Even Analysis

Businesses use systems to find when revenue equals costs.

Example:

A company's costs are $y = 500 + 2x$ (500 euros fixed + 2 euros per item) and revenue is $y = 5x$ (5 euros per item sold). The break-even point is where these lines intersect.

1Try It Yourself

Cost equation: $y = 200 + 3x$ . Revenue equation: $y = 8x$ .

At what quantity do costs equal revenue?

Step 1: Write the mathematical expression

Set the equations equal: $200 + 3x = 8x$

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 $y = 60x + 10$ and $y = 80x$ . Where do they meet?

2Try It Yourself

Train A: $y = 50x + 20$ (starts 20 km ahead, 50 km/h). Train B: $y = 70x$ (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: $50x + 20 = 70x$

Supply and Demand

Economists find equilibrium where supply equals demand.

Example:

Supply: $y = 2x + 10$ (price increases as quantity increases). Demand: $y = -x + 40$ (price decreases as quantity increases). The equilibrium is where these curves intersect.

3Try It Yourself

Supply: $y = 3x + 5$ . Demand: $y = -2x + 30$ .

What is the equilibrium price and quantity?

Step 1: Write the mathematical expression

Set supply equal to demand: $3x + 5 = -2x + 30$

Key Takeaways

1To solve a system by graphing, plot both lines and find where they intersect
2The intersection point $(x, y)$ 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

Graphing can be imprecise for non-integer solutions. In such cases, algebraic methods (substitution or elimination) give exact answers. However, you can estimate from the graph and verify algebraically.

Glossary

System of equations: Two or more equations with the same variables that are solved together
Solution of a system: An ordered pair $(x, y)$ 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)

Solving Systems by Graphing

Definition

Worked Examples

Common Mistakes

Interactive Visual

Linear Function Explorer

Interactive Sandbox

Interactive Grapher

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Break-Even Analysis

Meeting Point Problem

Supply and Demand

Key Takeaways

Frequently Asked Questions

Glossary

More in This Topic

Introduction to Systems of Equations

Systems of Equations (Substitution)

Substitution Method

Elimination Method