Algebra
Linear functions
Point-slope form
16 practice questions
2 video lessons
Theory + worked examples
Theory
Point-slope form writes a line from a point and the slope:
\[y-y_1=m(x-x_1),\]
where \((x_1,y_1)\) is a known point and \(m\) is the slope.
Best when you know a point and the slope — substitute and you're done.
A line through \((2,5)\) with slope \(3\).
Point-slope form.
Point-slope form:
\[y-y_1=m(x-x_1)\]
Distribute and solve for \(y\) to reach slope-intercept.
How to use point-slope form
- Identify the point \((x_1,y_1)\) and slope \(m\).
- Substitute into \(y-y_1=m(x-x_1)\).
- Distribute if a simplified form is needed.
- Solve for \(y\) to get slope-intercept form.
Example 1 — Write it
Write the line through \((2,5)\) with slope \(3\).
Solution
Use \(y-y_1=m(x-x_1)\).
| \(y-5\) | \(=\) | \(3(x-2)\) |
Example 2 — Convert to slope-intercept
Write \(y-5=3(x-2)\) as \(y=mx+b\).
Solution
Distribute and simplify.
| \(y-5\) | \(=\) | \(3x-6\) |
| \(y\) | \(=\) | \(3x-1\) |
Example 3 — Different point
Write the line through \((-1,4)\) with slope \(-2\).
Solution
Substitute the point and slope.
| \(y-4\) | \(=\) | \(-2(x+1)\) |
Example 4 — Why use it
When is point-slope form most convenient?
Solution
When you know one point and the slope — plug in directly.
Common pitfalls
Watch the signs: \((x-x_1)\) with a negative point becomes \((x+ \dots)\).
Substitute the point correctly for \(x_1\) and \(y_1\).
Distribute carefully when converting.
Frequently asked questions
What is point-slope form?
\(y-y_1=m(x-x_1)\), built from a point and slope.
When is point-slope form useful?
When you know a point and the slope.
How do you convert to slope-intercept?
Distribute and solve for \(y\).
Write the line through \((1,2)\) slope \(4\).
\(y-2=4(x-1)\).
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