Point-Slope and Slope-Intercept Forms

A Line Is an Equation

A line is not just a picture. It is an equation that every point on the line obeys. There are two forms worth knowing, and each is best in a different situation.

Slope-Intercept Form

The most common form packs two facts into one tidy equation:

$y = mx + b$

• $m$ is the slope
• $b$ is the y-intercept, the height where the line crosses the $y$-axis (at $x = 0$)

The animation shows what each letter does. $b$ slides the whole line up and down. $m$ tilts it. Change one and the other stays put.

You can read a line straight off this form:

$y = 2x + 3 \quad\Rightarrow\quad \text{slope } 2, \; \text{crosses the } y\text{-axis at } 3$

Point-Slope Form

Sometimes you do not know the $y$-intercept. You just know one point on the line and its slope. Point-slope form is built for exactly that:

$y - y_1 = m(x - x_1)$

Here $(x_1, y_1)$ is the known point and $m$ is the slope. It comes straight from the slope formula, rearranged so nothing is divided.

Example: a line through $(2, 5)$ with slope $3$:

$y - 5 = 3(x - 2)$

Converting Between Them

Point-slope is easy to write down; slope-intercept is easy to read. Expand one to get the other.

Starting from the example above:

Now the slope ($3$) and $y$-intercept ($-1$) are both in plain sight.

Which Form to Use

Write it in point-slope, then expand to slope-intercept whenever you want the numbers laid bare.