Examples · Graphing
Slope Intercept Form Examples: 9 Worked Problems
Nine problems, each adding one new difficulty. The last two are the ones that come up in exams and catch people out.
From two points
These slope intercept form examples all end in y = mx + b. The core skill is the same each time: find the slope, substitute one point, read off b, then check with the other point.
- m = (0 − 2) ÷ (4 − 0) = −0.5rise over run
- 2 = −0.5(0) + bsubstitute (0, 2)
- y = −0.5x + 2b = 2, because that point was already on the y-axis
- m = (11 − 3) ÷ (5 − 1) = 2the slope
- 3 = 2(1) + bsubstitute (1, 3)
- y = 2x + 1b = 1
- m = (−3 − 5) ÷ (2 − (−2))careful with the negatives
- m = −8 ÷ 4 = −2the run is 4, not 0
- y = −2x + 1substituting either point gives b = 1
From standard form
One rearrangement, with every term divided.
- 2y = −3x + 12subtract 3x
- y = −1.5x + 6divide all three terms by 2
- −y = −5x + 7subtract 5x
- y = 5x − 7multiply through by −1
Parallel and perpendicular
Both start from a known line and one point.
- m = 3parallel lines share a slope
- 1 = 3(2) + bsubstitute the point
- y = 3x − 5b = −5
- m = −½the negative reciprocal of 2
- 3 = −½(4) + bsubstitute the point
- y = −½x + 5b = 5
The two special cases
These are the examples worth practising, because a formula applied blindly gets both wrong.
- m = (4 − 4) ÷ (6 − 1) = 0no rise at all
- y = 4a horizontal line; the x term vanishes
- run = 3 − 3 = 0division by zero
- x = 3a vertical line — no slope-intercept form exists
Example 9 has no y = mx + b answer at all, and saying so is the correct response. A calculator that returns infinity here is giving a wrong answer rather than a useful one.
Two more, from a graph and a table
Real questions do not always hand you two tidy coordinates.
- x: 0, 1, 2, 3 y: 5, 8, 11, 14the table
- m = 3y rises by 3 each time x rises by 1
- y = 3x + 5b is the y value when x = 0, read straight off
- b = −2read the y-intercept off the axis
- m = (4 − (−2)) ÷ 3 = 2rise over run to the second point
- y = 2x − 2no substitution needed
A table with evenly spaced x values is the easiest case of all: the constant difference in y is the slope, and the value at x = 0 is b.
Checking each answer
Substitute both original points back into the final equation. Both must satisfy it, and checking the point you did not use to find b is the check that actually tests anything.
| Example | Equation | Check with the unused point |
|---|---|---|
| 2 | y = 2x + 1 | 2(5) + 1 = 11 ✓ |
| 3 | y = −2x + 1 | −2(2) + 1 = −3 ✓ |
| 4 | y = −1.5x + 6 | 3(0) + 2(6) = 12 ✓ |
Questions about slope intercept form examples
How do I find the equation of a line from two points?
Find the slope with (y₂ − y₁)/(x₂ − x₁), substitute either point into y = mx + b, and solve for b. Then check with the other point.
What is the equation of a horizontal line?
y = c, where c is the shared y value. The slope is zero, so the x term disappears entirely.
What if the two points have the same x value?
The line is vertical, its slope is undefined, and it is written x = a. There is no slope-intercept form for it.
How do I find a perpendicular line?
Take the negative reciprocal of the original slope, then substitute the given point to find the new b.