4.2Solving Systems by Graphing
Graph both equations on the same coordinate plane. The point where the lines cross is the solution to the system.
Why This Matters
Graphing systems gives you a visual understanding of where two relationships intersect — a concept used in economics to find market equilibrium and in Physics to find when two objects meet. It builds intuition for all future graphing work.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How can graphing two linear equations on the same coordinate plane reveal the solution to a system, and what does the graph look like when there is no solution or infinitely many solutions?
Lesson Overview
To solve a system by graphing, graph both equations on the same coordinate plane and look for the point where they intersect. That intersection point (x, y) is the solution. If the lines are parallel they never meet — no solution. If both equations describe the same line, every point on the line is a solution — infinitely many solutions. Always verify your graphical answer by substituting back into both original equations.
One Solution — Lines Intersect at (1, 2)
No Solution — Parallel Lines
Infinitely Many Solutions — Coincident Lines
Three possible outcomes when graphing a system
Step-by-Step Graphing Method
- Rewrite both equations in slope-intercept form (y = mx + b).
- Plot the y-intercept of the first equation on the y-axis.
- Use the slope (rise/run) to plot a second point, then draw the line.
- Repeat steps 2–3 for the second equation.
- Identify the intersection point — or state "no solution" / "infinitely many."
- Verify by substituting the intersection point into both original equations.
y = ²⁄₃x − 1 · Plot intercept, then rise/run
Graphing from a table of values
Worked Examples
Intersection = solution (2, 3)
Intersection = solution (1, 1)
Solve the system by graphing: y = x + 1 and y = −x + 5
Equation 1: y = x + 1 → slope m = 1, y-intercept b = 1. Plot (0, 1); use slope to plot (1, 2) and (2, 3).
Equation 2: y = −x + 5 → slope m = −1, y-intercept b = 5. Plot (0, 5); use slope to plot (1, 4) and (2, 3).
Both lines pass through (2, 3) → that is the intersection.
Verify: Eq 1: 3 = 2 + 1 = 3 ✓ Eq 2: 3 = −2 + 5 = 3 ✓
Solution: (2, 3)
Solve the system by graphing: y = 2x + 3 and y = 2x − 2
Equation 1: slope 2, y-intercept 3.
Equation 2: slope 2, y-intercept −2.
Same slope (m = 2), different y-intercepts → parallel lines.
Parallel lines never intersect.
No Solution — Parallel Lines (m = 2)
Solve the system by graphing: y = x + 2 and 2y = 2x + 4
Rewrite Equation 2: divide both sides by 2 → y = x + 2.
Both equations are identical → same line.
Every point on the line is a solution.
Infinitely Many Solutions — Same Line
Solve the system by graphing: x + y = 4 and 2x − y = 2
Rewrite Equation 1: y = −x + 4 (slope −1, y-intercept 4).
Rewrite Equation 2: y = 2x − 2 (slope 2, y-intercept −2).
Graph both lines. Intersection at (2, 2).
Verify: Eq 1: 2+2 = 4 ✓ Eq 2: 2(2)−2 = 2 ✓
Rewrite in slope-intercept form first → (2, 2)
Solve the system by graphing: y = −2x + 4 and y = x + 1
Equation 1: slope −2, y-intercept 4. Plot (0, 4), (1, 2), (2, 0).
Equation 2: slope 1, y-intercept 1. Plot (0, 1), (1, 2), (2, 3).
Intersection at (1, 2).
Verify: Eq 1: 2 = −2(1)+4 = 2 ✓ Eq 2: 2 = 1+1 = 2 ✓
Solution: (1, 2)
Guided Practice
Guided Practice Video: Solving Systems by Graphing
Watch the guided practice walkthrough for solving systems by graphing, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Graph the system y = x + 3 and y = −x + 1. Identify the solution.
Hint: Plot the y-intercept of each line, then use the slope to find a second point. Where do the lines cross?
Graph the system y = 2x − 4 and y = x − 1. Identify the solution and verify it.
Hint: Equation 1: y-intercept −4, slope 2. Equation 2: y-intercept −1, slope 1. Find the intersection, then substitute back.
Graph the system y = 3x + 1 and y = 3x − 2. How many solutions? What type of system?
Hint: Compare the slopes first. What does it mean when two lines have the same slope?
Rewrite x − y = 2 and 2x + y = 7 in slope-intercept form, then graph and solve.
Hint: Solve each equation for y. Then graph using slope and y-intercept.
Graph y = −½x + 3 and y = x. Find the solution and verify.
Hint: For y = x, the slope is 1 and y-intercept is 0. Plot both lines carefully.
Key Vocabulary
Point of Intersection
The ordered pair (x, y) where two lines cross on a graph — this is the solution to the system.
Slope-Intercept Form
y = mx + b, where m is the slope and b is the y-intercept. The most useful form for graphing.
Slope (m)
The rate of change of a line; rise over run. Determines how steep the line is and in which direction it tilts.
y-Intercept (b)
The point where a line crosses the y-axis; the value of y when x = 0.
Parallel Lines
Lines with the same slope but different y-intercepts. They never intersect → no solution.
Coincident Lines
Two equations that produce the exact same line. Every point is a solution → infinitely many solutions.
Table of Values
A method of graphing by choosing x-values, computing y-values, and plotting the resulting ordered pairs.
Verification
Substituting the solution back into both original equations to confirm it satisfies both.
Interactive Multiple Choice Practice
Interactive Practice — 5 Questions
What is the solution to the system y = x + 1 and y = −x + 5?
Two lines have the same slope but different y-intercepts. What is the solution to the system?
What is the first step when solving a system by graphing if equations are in standard form?
A student graphs y = 2x + 1 and y = 2x − 3. What type of system is this?
After graphing a system, you find the lines intersect at (3, −1). How do you verify this solution?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Solve by graphing: y = x + 2 and y = −x + 4.
Solve by graphing: y = 2x + 1 and y = x + 3.
Solve by graphing: y = 3x − 3 and y = x + 1.
Solve by graphing: y = 4x − 2 and y = 4x + 3. Classify the system.
Rewrite and solve by graphing: x + y = 5 and x − y = 1.
Common Mistakes
Reading the intersection point incorrectly — estimating a non-integer answer when the actual solution is a whole number.
Graph carefully using slope and y-intercept. If the intersection looks like a fraction, double-check your lines.
Forgetting to rewrite equations in slope-intercept form before graphing.
Convert to y = mx + b first. Identify slope and y-intercept clearly before plotting any points.
Plotting the y-intercept on the x-axis or misapplying slope direction.
The y-intercept goes on the y-axis at (0, b). Slope = rise/run — rise is vertical, run is horizontal.
Not verifying the graphed intersection by substituting back into both equations.
Always check: substitute the intersection point into both original equations to confirm it satisfies both.
Math Tips
Always rewrite equations in slope-intercept form (y = mx + b) before graphing — it makes plotting fast and accurate.
Use two different colors for the two lines so the intersection point is easy to spot.
Plot at least 3 points per line: the y-intercept plus two points using the slope (rise/run).
If the lines have the same slope, they are parallel — no need to graph fully, the answer is "no solution."
Always verify your graphical solution by substituting the intersection point into both original equations.