01 / 21

Solving Systems
by Graphing

Learn to solve systems of linear equations by graphing two lines and finding where they intersect: the point that makes both equations true.

Pre-Algebra

What Is a System of Equations?

A system of equations is a set of two or more equations that share the same variables. When we "solve" a system, we find values for those variables that make every equation true at the same time.

Why Solve by Graphing?

Each linear equation represents a straight line on the coordinate plane. Every point on that line is a solution to that one equation. When two lines cross, the intersection point is a solution to both equations.

Visual
You can literally see where the solution lives on the graph.
Intuitive
Great first method: builds understanding before algebraic techniques.
Real-World
Models situations where two quantities meet: cost vs. revenue, speed, etc.

The Intersection Is the Solution

x y 0 1 2 3 4 5 -1 -2 -3 1 2 3 4 -1 (2, 3)
\( y = x + 1 \)
\( y = -x + 5 \)

Three Possible Outcomes

One Solution
Lines cross at exactly one point. Different slopes. The system is consistent and independent.
No Solution
Lines are parallel: same slope, different y-intercepts. They never meet. The system is inconsistent.
Infinitely Many
Lines are identical: same slope and same y-intercept. Every point on the line is a solution.

Quick Review: Slope-Intercept Form

To graph a line, we use slope-intercept form:

Step 1
Plot the y-intercept \( b \) on the y-axis.
Step 2
From that point, use the slope \( m = \frac{\text{rise}}{\text{run}} \) to plot a second point.
Step 3
Draw a straight line through both points and extend it.

One Equation vs. a System

One Equation System of Two
Graph One line Two lines
Solutions Infinitely many \( (x,y) \) pairs on the line Usually one \( (x,y) \) pair (the intersection)
Goal Describe or graph the line Find where lines meet
Check Plug into one equation Plug into both equations

How to Solve by Graphing

1
Write both equations in slope-intercept form \( y = mx + b \).
2
Graph the first line. Plot the y-intercept, then use the slope to find a second point. Draw the line.
3
Graph the second line on the same coordinate plane.
4
Identify the intersection point \( (x, y) \). If the lines don't cross, the system has no solution.
5
Check your answer by substituting into both original equations.

Example 1: Finding the Intersection

0 1 2 3 4 5 -1 -2 -3 1 2 3 4 -1 -2 x y (2, 3)
\( y = x + 1 \)
\( y = -x + 5 \)

Example 2: No Solution

1 2 3 -1 -2 1 2 3 -1 -2
\( y = 2x + 1 \)
\( y = 2x - 3 \)
 Parallel: no intersection

Example 3: Real-World Context

Scenario: You and a friend start a savings challenge. You already have $20 and save $5 per week. Your friend starts with $50 and saves $2 per week. When will you have the same amount?

Example 3: The Graph

0 2 4 6 8 10 12 20 40 60 80 Weeks Dollars (10, $70) You Friend

What If Equations Aren't in \( y = mx + b \)?

Sometimes equations come in standard form: \( Ax + By = C \). You need to rearrange first.

When the Intersection Isn't Exact

Graphing is great for understanding, but it has a limitation: if the intersection point has non-integer coordinates (like \( (1.5, 3.7) \)), it can be hard to read precisely from a hand-drawn graph.

Solution
Estimate from the graph, then verify algebraically by solving the system with substitution or elimination.
Technology
Graphing calculators and apps like Desmos can find exact intersection points for you.

Common Mistakes to Avoid

✕ Wrong Slope Direction
A negative slope goes down from left to right. Confusing positive and negative slopes shifts the entire line.
✕ Forgetting to Check
Always plug the intersection point back into both equations. If it fails either one, re-graph carefully.
✕ Reading the Graph Wrong
Misreading the scale or swapping \( x \) and \( y \) coordinates. Double-check which axis is which.

Summary: System Outcomes

What You See Lines Cross Lines Parallel Lines Overlap
Slopes Different Same Same
Y-Intercepts Any Different Same
Solutions Exactly one None Infinitely many
Term Consistent, Independent Inconsistent Consistent, Dependent

Question 1 of 5

What does the intersection point of two graphed lines represent?

Question 2 of 5

The system \( y = 3x + 2 \) and \( y = 3x - 4 \) has:

Question 3 of 5

Solve by graphing: \( y = x + 2 \) and \( y = -2x + 8 \). What is the solution?

Question 4 of 5

A gym charges $25 per month plus a $50 joining fee. Another gym charges $35 per month with no joining fee. After how many months do both cost the same?

Question 5 of 5

If two lines have the same slope and the same y-intercept, how many solutions does the system have?