3.6Graphing Linear Equations
A linear equation in two variables describes an infinite set of points that form a perfectly straight line. Learn to graph lines using tables of values, intercepts, and slope.
Why This Matters
Graphing linear equations is the visual foundation of algebra — every linear relationship in science, economics, and engineering can be represented as a line on a coordinate plane.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
What are the three methods for graphing a linear equation, and when is each method most efficient?
Lesson Overview
A linear equation in two variables has infinitely many solutions — each solution is an ordered pair (x, y) that makes the equation true. When plotted, all solutions lie on a straight line. There are three efficient methods for graphing: (1) Table of values — substitute several x-values, compute y, then plot and connect the points; (2) Intercepts method — find the x-intercept (set y = 0) and the y-intercept (set x = 0), then draw the line through those two points; (3) Slope-intercept method — rewrite the equation as y = mx + b, plot the y-intercept b, then use slope m to find additional points.
Three Methods at a Glance
1. Table of Values
Choose x-values, substitute into the equation, compute y. Plot all points and connect.
Best when: equation is solved for y
2. Intercepts
Set y = 0 to find x-intercept. Set x = 0 to find y-intercept. Connect the two points.
Best when: standard form Ax + By = C
3. Slope-Intercept
Plot b on y-axis. Use slope m = rise/run to find more points.
Best when: y = mx + b form
Three Methods — Visual Examples
y = 2x − 1 (table method)
Substitute x = −2, 0, 2 into y = 2x − 1
3x + 2y = 6 (intercepts)
x-int: (2,0) when y=0; y-int: (0,3) when x=0
y = −(1/2)x + 3 (slope)
Plot b=3, then go down 1, right 2 for each point
Worked Examples
Graph y = 2x − 1 using a table of values.
Choose x-values: x = −2, −1, 0, 1, 2.
Substitute each into y = 2x − 1:
x = −2: y = 2(−2) − 1 = −5 → (−2, −5)
x = −1: y = 2(−1) − 1 = −3 → (−1, −3)
x = 0: y = 2(0) − 1 = −1 → (0, −1)
x = 1: y = 2(1) − 1 = 1 → (1, 1)
x = 2: y = 2(2) − 1 = 3 → (2, 3)
Plot all five points on the coordinate plane.
Draw a straight line through all five points.
Example 1 — y = 2x − 1
Graph 3x + 2y = 6 using the intercepts method.
Find the x-intercept: set y = 0.
3x + 2(0) = 6 → 3x = 6 → x = 2. x-intercept: (2, 0).
Find the y-intercept: set x = 0.
3(0) + 2y = 6 → 2y = 6 → y = 3. y-intercept: (0, 3).
Plot (2, 0) and (0, 3) on the coordinate plane.
Draw a straight line through both intercept points.
Check: pick a third point, e.g. x = 4: 3(4) + 2y = 6 → y = −3. Plot (4, −3) — it should be on the line.
Example 2 — 3x + 2y = 6
Graph y = −(1/2)x + 3 using the slope-intercept method.
Identify slope m = −1/2 and y-intercept b = 3.
Plot the y-intercept: (0, 3).
Use slope −1/2: from (0,3) go down 1, right 2 → (2, 2).
Apply slope again: from (2,2) go down 1, right 2 → (4, 1).
Apply slope again: from (4,1) go down 1, right 2 → (6, 0).
Draw a straight line through all plotted points.
Graph x = 4.
The equation x = 4 means every point has x-coordinate equal to 4.
y can be any value: (4, −3), (4, 0), (4, 2), (4, 5) are all solutions.
Plot several points with x = 4 and different y-values.
Draw a vertical line through x = 4.
Note: this is NOT a function — it fails the vertical line test.
The slope of a vertical line is undefined.
Example 4 — x = 4 (vertical line)
Graph y = −2.
The equation y = −2 means every point has y-coordinate equal to −2.
x can be any value: (−3, −2), (0, −2), (4, −2) are all solutions.
Plot several points with y = −2 and different x-values.
Draw a horizontal line through y = −2.
The slope of a horizontal line is 0.
Guided Practice
Guided Practice Video: Graphing Linear Equations
Watch the guided practice walkthrough for graphing linear equations, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Graph y = 3x − 2 using a table of values for x = −1, 0, 1, 2.
Hint: Substitute each x: y = 3(−1)−2 = −5; y = 3(0)−2 = −2; y = 3(1)−2 = 1; y = 3(2)−2 = 4. Points: (−1,−5), (0,−2), (1,1), (2,4).
Graph 2x + 3y = 12 using the intercepts method.
Hint: x-intercept: set y=0 → 2x=12 → x=6, so (6,0). y-intercept: set x=0 → 3y=12 → y=4, so (0,4). Draw line through (6,0) and (0,4).
Graph y = (2/3)x − 1 using the slope-intercept method.
Hint: b = −1, so start at (0,−1). Slope 2/3: go up 2, right 3 → (3,1). Go up 2, right 3 again → (6,3).
Graph x = −3. What type of line is it? What is its slope?
Hint: Vertical line at x = −3. Plot (−3,−2), (−3,0), (−3,3). Slope is undefined.
Graph y = 5. What type of line is it? What is its slope?
Hint: Horizontal line at y = 5. Plot (−2,5), (0,5), (3,5). Slope is 0.
Key Vocabulary
x-intercept
The point where a line crosses the x-axis. At this point, y = 0.
Example: For 3x + 2y = 6: set y=0 → x=2. x-intercept is (2,0).
y-intercept
The point where a line crosses the y-axis. At this point, x = 0.
Example: For 3x + 2y = 6: set x=0 → y=3. y-intercept is (0,3).
Table of Values
A method of graphing by substituting x-values into an equation to find corresponding y-values, then plotting the ordered pairs.
Example: For y = 2x−1: x=0 gives y=−1, x=1 gives y=1, etc.
Linear Equation
An equation whose graph is a straight line. In two variables, it has the form Ax + By = C.
Example: y = 2x − 1, 3x + 2y = 6, and x = 4 are all linear equations.
Solution of a Linear Equation
An ordered pair (x, y) that makes the equation true. Every point on the graph is a solution.
Example: (1, 1) is a solution of y = 2x−1 because 1 = 2(1)−1 = 1 ✓
Interactive Practice — 5 Questions
Which ordered pair is a solution of y = 2x − 1?
What is the x-intercept of 3x + 2y = 6?
Which method is most efficient for graphing 4x − 3y = 12?
What is the slope of the line x = 7?
A line has y-intercept (0, −3) and slope 2. Which equation describes it?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Graph y = −x + 4 using a table of values for x = −2, −1, 0, 1, 2.
Graph 2x + y = 6 using the intercepts method. Identify both intercepts.
Graph y = (3/4)x − 2 using the slope-intercept method.
Graph y = 3 and x = −5 on the same coordinate plane. Describe each line.
Challenge: Graph 5x − 2y = 10 using all three methods (table, intercepts, slope-intercept). Verify that all three methods produce the same line.
ChallengeCommon Mistakes
Plotting (x, y) as (y, x) — reversing the coordinates.
Always plot (x, y): x is the horizontal position, y is the vertical position. The x-coordinate comes first.
Using only two points from a table and assuming the line is correct without checking a third point.
Always plot at least three points. If they don't line up, recheck your substitution — a mistake in one point is easy to spot.
Confusing x-intercept and y-intercept when using the intercepts method.
x-intercept: set y = 0 (the line crosses the x-axis). y-intercept: set x = 0 (the line crosses the y-axis).
Thinking x = 4 and y = 4 describe the same line.
x = 4 is a vertical line (all points have x = 4). y = 4 is a horizontal line (all points have y = 4). They are perpendicular to each other.
Math Tips
Always use at least three points when graphing — two points define a line, but a third point confirms you made no arithmetic errors.
For the intercepts method, the x-intercept always has y = 0 and the y-intercept always has x = 0. A quick memory trick: the intercept is named after the axis it touches.
When using slope-intercept form, write the slope as a fraction (rise/run) even if it is a whole number — e.g., m = 3 becomes 3/1 (up 3, right 1).
Vertical lines (x = a) have undefined slope and are NOT functions. Horizontal lines (y = b) have slope 0 and ARE functions.