What Is Slope Intercept Form?

Slope-intercept form is a way of writing the equation of a straight line as y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis).

It's the most useful form of a linear equation because you can read both key facts about the line straight off the equation, without doing any work.

Breaking down y = mx + b:

• y — the output, or vertical coordinate.
• m — the slope: how steep the line is, calculated as rise over run (change in y ÷ change in x).
• x — the input, or horizontal coordinate.
• b — the y-intercept: the y-value where the line crosses the y-axis, i.e. the point (0, b).

Example: in y = 3x + 2

• The slope is 3 — the line rises 3 units for every 1 unit it moves right.
• The y-intercept is 2 — the line passes through (0, 2).

Reading the slope:

• Positive m → the line goes up from left to right.
• Negative m → the line goes down from left to right.
• m = 0 → a horizontal line (y = b).
• A larger absolute value of m means a steeper line.
• A vertical line cannot be written in slope-intercept form, because its slope is undefined. It's written as x = a constant.

Watch the signs. In y = 2x − 5, the y-intercept is −5, not 5, since the form is + b. And in y = −x + 4, the slope is −1 (an invisible coefficient of 1).

How to graph a line from slope-intercept form — three steps:

1. Plot the y-intercept at (0, b).
2. Use the slope as rise over run from that point. For m = 3, treat it as 3/1: go up 3, right 1, and plot a second point. For m = −2/3: go down 2, right 3.
3. Draw a straight line through the two points.

How to write an equation in slope-intercept form:

From a slope and a point: substitute the slope for m and the point's coordinates for x and y, then solve for b. With slope 4 through the point (2, 5): 5 = 4(2) + b → 5 = 8 + b → b = −3, so y = 4x − 3.

From two points: first find the slope with m = (y₂ − y₁) / (x₂ − x₁), then use either point to solve for b as above.

From standard form: standard form is Ax + By = C. Just solve for y. Given 2x + 4y = 8: subtract 2x to get 4y = −2x + 8, then divide everything by 4 → y = −½x + 2.

How it compares to the other forms of a line:

Why it's useful in real problems: any situation with a starting amount and a constant rate of change fits this form. A phone plan costing $30 a month plus a $50 activation fee is y = 30x + 50, where b is the one-off starting cost and m is the rate per month.