Key takeaway: Point-slope form writes a line’s equation as y minus y1 equals m times x minus x1, using the slope m and any one point on the line.
Grade 9 – Algebra 1A-CED.A.2Functions and Graphs
Definition of Point-Slope Form
Point-slope form is the fastest way to write the equation of a line when you know its slope and one point it passes through. You do not need the y-intercept first.
The idea comes from the slope itself. For any other point \((x, y)\) on the line, the rise divided by the run to the known point equals \(m\); multiplying both sides by the run gives the equation. The form also works when you know two points: compute the slope, then use either point. Because it keeps the point visible, the form is a handy step before simplifying to slope-intercept or standard form.
Formula and notation
With slope \(m\) and a known point \((x_1, y_1)\):
\[y – y_1 = m(x – x_1)\]
The subtraction signs are part of the pattern. A known point with a negative coordinate turns them into addition, for example \(y – 7 = m(x + 2)\) for the point \((-2, 7)\).
Key facts
- The point in the equation is a point on the line, with its signs reversed in the parentheses.
- Different points on the same line give different-looking but equivalent equations.
- Distributing m and adding y1 converts the equation to slope-intercept form.
- Vertical lines cannot use this form because their slope is undefined.
Point-Slope Form example
A line has slope \(-\frac{3}{2}\) and passes through \((-2, 7)\). Write it in point-slope form, then in slope-intercept form.
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Substitute \(m = -\frac{3}{2}\), \(x_1 = -2\), \(y_1 = 7\): \(y – 7 = -\frac{3}{2}(x + 2)\).
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Distribute: \(y – 7 = -\frac{3}{2}x – 3\).
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Add 7 to both sides: \(y = -\frac{3}{2}x + 4\). Check at \(x = -2\): \(3 + 4 = 7\).
Point-slope form: \(y – 7 = -\frac{3}{2}(x + 2)\); slope-intercept form: \(y = -\frac{3}{2}x + 4\).
Common mistake
Watch out
Using the point \((-2, 7)\), Eli writes \(y – 7 = -\frac{3}{2}(x – 2)\).
Why it is wrong
The pattern is \(x – x_1\), and \(x_1 = -2\), so \(x – (-2) = x + 2\). Writing \(x – 2\) describes a line through \((2, 7)\) instead. Test the equation by putting the point back in: both sides must give 0.
Related terms and lessons
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