Point-Slope Form
Line through a point with a given slope.
Formula
Variables
Example
Line through (2, 3) with slope 4: y − 3 = 4(x − 2), i.e. y = 4x − 5.
How it works
This formula gives the equation of a straight line when you know one point on the line and its slope. It works because slope measures the rate of change, so the vertical change from the known point to any other point must equal the slope times the horizontal change. Rearranging that relationship produces the point-slope form.
When to use it
Use this when you know one point and the slope and need the equation of a line quickly. It is especially useful in algebra, coordinate geometry, and physics problems involving constant rate of change.
Worked example, step by step
Find the equation of the line through (-1, 5) with slope -2. Start with the point-slope form: y - y1 = m(x - x1) Substitute m = -2, x1 = -1, y1 = 5: y - 5 = -2(x - (-1)) Simplify inside the parentheses: y - 5 = -2(x + 1) Distribute: y - 5 = -2x - 2 Add 5 to both sides: y = -2x + 3 Final answer: y = -2x + 3
Common mistakes to avoid
- Using the slope with the wrong sign, especially when the slope is negative
- Substituting x1 and y1 into the wrong places in the formula
- Stopping at point-slope form when the problem asks for slope-intercept form
Did You Know?
Point-slope form is the fastest way to write a line's equation when you know one point and the slope.
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