Point-Slope Form Examples: 10 Solved Problems
The fastest way to get comfortable with point-slope form is to watch it work over and over. Below are ten examples, each solved step by step, starting easy and building up to two-point and special cases. The formula for every one is y − y₁ = m(x − x₁).
Quick Refresher: The Point-Slope Formula
Before the examples, here’s the tool we’ll use each time. If you know a point (x₁, y₁) and a slope m, the line is:
y − y₁ = m(x − x₁)
Examples From a Point and Slope
Example 1: Positive Slope
Point (2, 3), slope 4. Substitute: y − 3 = 4(x − 2). Nothing tricky here, since both numbers are positive.
Example 2: Negative Slope
Point (1, 5), slope −2. The slope keeps its sign: y − 5 = −2(x − 1).
Example 3: Fractional Slope
Point (4, 1), slope 3/4. Fractions are fine as a slope: y − 1 = (3/4)(x − 4). Don’t convert to a decimal, because the fraction stays exact.
Example 4: Negative Coordinate
Point (3, −2), slope 5. Watch the sign: y − (−2) becomes plus. y + 2 = 5(x − 3).
Examples From Two Points
Example 5: Two Positive Points
Points (1, 2) and (4, 8). Slope first: m = (8 − 2) / (4 − 1) = 2. Using (1, 2): y − 2 = 2(x − 1).
Example 6: A Point With Negatives
Points (−2, 1) and (2, 9). Slope: m = (9 − 1) / (2 − (−2)) = 8/4 = 2. Using (−2, 1): y − 1 = 2(x + 2).
Special Cases
Example 7: Zero Slope (Horizontal Line)
Point (3, 4), slope 0. Substitute: y − 4 = 0(x − 3), which simplifies to y = 4. A zero slope always gives a flat, horizontal line.
Example 8: Undefined Slope (Vertical Line)
Points (2, 1) and (2, 7). Both x-values are 2, so the slope is undefined and point-slope form doesn’t apply. The line is simply x = 2.
Examples That Convert to Other Forms
Example 9: Convert to Slope-Intercept
Start with y + 2 = 5(x − 3) from Example 4. Distribute: y + 2 = 5x − 15. Subtract 2: y = 5x − 17.
Example 10: Word-Problem Style
A gym charges a $30 signup fee plus $20 a month. At month 0 the cost is $30, and the rate is 20, so a point is (0, 30) with slope 20. That gives y − 30 = 20(x − 0), or y = 20x + 30. Now you can predict the cost for any month.
Ready to practice on your own? Grab the free worksheet, review the three-step method, or check answers with the calculator.