Point-Slope Form
In short: Point-slope form builds a line’s equation from one known point \((x_1, y_1)\) and the slope m: \(y – y_1 = m(x – x_1)\).
Algebra 1Functions and GraphsA-CED.A.2
What is Point-Slope Form?
When a problem hands you a single location on a line plus its steepness, point-slope form lets you write the equation right away, without hunting for the y-intercept first.
The pattern is \(y – y_1 = m(x – x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the known point. Read it as: the vertical change from the known point equals the slope times the horizontal change. Notice the signs flip inside the parentheses: a point \((2, -1)\) gives \(y + 1 = m(x – 2)\).
How to spot it
- The facts supplied are one location on the line and its slope, and the target is an equation.
- Two locations are supplied, so you compute the slope first and then reuse either location.
- The question wants an equation but never mentions where the line meets the y-axis.
- A new line must run parallel or perpendicular to a given one while passing through a stated point.
The method
- Find the slope m, using \(\frac{y_2 – y_1}{x_2 – x_1}\) if you were given two points.
- Choose one point \((x_1, y_1)\) on the line.
- Substitute into \(y – y_1 = m(x – x_1)\), being careful with minus signs when a coordinate is negative.
- If asked, distribute m and add \(y_1\) to both sides to reach slope-intercept form.
Point-Slope Form: solved example
Problem
A zip line cable runs through the point \((-3, 5)\) on a map grid and has slope \(-\frac{2}{3}\). Give its equation first in point-slope form, then in slope-intercept form.
Solution
With \((x_1, y_1) = (-3, 5)\) and \(m = -\frac{2}{3}\): \(y – 5 = -\frac{2}{3}(x – (-3))\), which is \(y – 5 = -\frac{2}{3}(x + 3)\).
Distribute: \(y – 5 = -\frac{2}{3}x – 2\). Add 5: \(y = -\frac{2}{3}x + 3\).
Try it
Question 1
A line with slope 5 goes through \((4, 7)\). Give its point-slope equation.
Show the answer
\(y – 7 = 5(x – 4)\).
Question 2
Find an equation for the line joining \((1, 2)\) and \((5, 10)\).
Show the answer
The slope is \(\frac{10 – 2}{5 – 1} = 2\), so \(y – 2 = 2(x – 1)\).
Point-Slope Form vs. Slope-Intercept Form
Slope-intercept form, \(y = mx + b\), needs the y-intercept. This form needs any point at all, so it is the quick choice when the given point is not on the y-axis. Both describe the same line, and algebra converts one into the other.
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Related words
- Slope-Intercept Form
- Standard Form of a Linear Equation
- Slope
- Slope Formula
- Linear Equation
- Parallel Lines
Whole dictionary: A–Z · Algebra 1 · Functions and Graphs