Key idea
The difference between point slope form and slope intercept form is mainly about which information is emphasized: a known point on the line versus the y-intercept. Both describe the same family of non-vertical lines and can be converted by algebraic rearrangement.
Definitions
Point-slope form (uses a known point and slope):
\[ y - y_1 = m\cdot(x - x_1) \]
Slope-intercept form (isolates \(y\) and shows intercept directly):
\[ y = m\cdot x + b \]
Here \(m\) is the slope, \((x_1,y_1)\) is a known point on the line, and \(b\) is the y-intercept (the value of \(y\) when \(x=0\)).
Comparison table
| Feature | Point-slope form \(y - y_1 = m\cdot(x - x_1)\) | Slope-intercept form \(y = m\cdot x + b\) |
|---|---|---|
| What is shown immediately? | A specific point \((x_1,y_1)\) and the slope \(m\) | The slope \(m\) and the y-intercept \(b\) |
| Best used when | A point on the line and the slope are given | Graphing quickly from intercept and slope, or reading intercept directly |
| Graphing starting point | Start at \((x_1,y_1)\), then apply rise/run from \(m\) | Start at \((0,b)\), then apply rise/run from \(m\) |
| Common mistake | Sign errors in \(x - x_1\) or \(y - y_1\) | Forgetting to isolate \(y\) correctly |
| Limitation | Does not show \(b\) unless converted | Cannot represent vertical lines (no finite slope) |
How to convert point-slope form to slope-intercept form
Converting is an algebraic “expand and isolate \(y\)” procedure. Consider the concrete example:
\[ y - (-1) = \frac{3}{2}\cdot(x - 2) \]
- Rewrite the left side: \[ y + 1 = \frac{3}{2}\cdot(x - 2) \]
- Distribute \(\frac{3}{2}\) across the parentheses: \[ y + 1 = \frac{3}{2}\cdot x - \frac{3}{2}\cdot 2 \] \[ y + 1 = \frac{3}{2}\cdot x - 3 \]
- Subtract \(1\) from both sides to isolate \(y\): \[ y = \frac{3}{2}\cdot x - 4 \]
Converted slope-intercept form: \(\;y = \frac{3}{2}\cdot x - 4\)
The y-intercept is \(b=-4\), so the line crosses the y-axis at \((0,-4)\).
How to convert slope-intercept form to point-slope form
Start with \(y = m\cdot x + b\). Choose any point on the line, then substitute that point into \((x_1,y_1)\). A convenient choice is the y-intercept point \((0,b)\), which always lies on a non-vertical line.
For the example \(y = \frac{3}{2}\cdot x - 4\), the point \((0,-4)\) lies on the line. Substituting gives:
\[ y - (-4) = \frac{3}{2}\cdot(x - 0) \]
\[ y + 4 = \frac{3}{2}\cdot x \]
Visualization: point, slope, and the same line in two forms
Important special case
Vertical lines have undefined slope and cannot be written in slope-intercept form. A vertical line through \(x=c\) is written as \[ x = c. \] Point-slope form also assumes a finite slope \(m\), so it is not used for vertical lines.