Skill: slope-intercept

Slope-Intercept Form

Every non-vertical line has a name tag: y = mx + b. The m tells you the slope, the b tells you where the line crosses the y-axis. Learn to read both at a glance, graph any line in seconds, and write the equation from a graph or two points.

1 Reading y = mx + b

In y = mx + b, the number in front of x is the slope m and the lonely constant is the y-intercept b — the point (0, b) where the line crosses the y-axis.

Slope-intercept form

y = mx + b

m = slope (rise over run). b = y-intercept: the line passes through (0, b). b is not the x-intercept — it lives on the y-axis.

y = 2x − 3: start at b = −3 on the y-axis, then climb 2 for every 1 across (m = 2).

Graphing recipe: plot (0, b), then use the slope as rise/run to find a second point, and draw the straight line through both. Negative slope? Fall as you move right.

2 Worked examples

Read, evaluate, write, and build from two points.

Example 1 Identify m and b

In y = (2/3)x − 4, the slope is m = 2/3 and the y-intercept is b = −4, the point (0, −4).

Example 2 Evaluate

If y = 2x + 3, find y when x = 4.

y = 2(4) + 3 = 11 — the point (4, 11) is on the line.

Example 3 Write the equation

Write the equation with slope −1 and y-intercept 5.

y = −x + 5. A slope of −1 is written −x, not −1x.

Example 4 From two points

Write the equation through (1, 2) and (4, 8).

Slope: m = (8 − 2)/(4 − 1) = 2. Plug into y = mx + b with (1, 2): 2 = 2(1) + b gives b = 0.

So y = 2x. A zero intercept means the line passes through the origin.

3 Common mistakes

Three traps, each with the wrong version and the fix.

1. Reading b as the x-intercept

Wrong: in y = 2x − 3, the x-intercept is −3. Right: b = −3 is the y-intercept (0, −3). The x-intercept needs solving: 0 = 2x − 3 gives x = 3/2.

2. Graphing a negative slope backwards

Wrong: y = −2x + 1 drawn climbing to the right. Right: negative slope falls left to right: from (0, 1), go down 2 for each 1 across.

3. Forgetting b when writing from a graph

Wrong: a line with slope 2 through (0, 5) written as y = 2x. Right: y = 2x + 5 — b shifts the whole line up.

4 Quick checks

Try these yourself, then reveal the answer.

Identify m and b in y = −4x + 7.

m = −4, b = 7.

Write the equation with slope 3/2 and y-intercept −1.

y = (3/2)x − 1.

Find the x-intercept of y = 2x − 6.

Set y = 0: 0 = 2x − 6 gives x = 3.

5 Key points

Remember

y = mx + b: m is the slope, b is the y-intercept (0, b).
Graph: plot (0, b), then step rise/run from the slope.
From two points: slope first, then solve for b with either point.
The x-intercept is where y = 0: solve 0 = mx + b.
Horizontal lines: y = b (m = 0). Vertical lines: x = a — not slope-intercept form.

Key vocabulary

Slope-intercept form

y = mx + b, with slope m and y-intercept b built in.

y-intercept

Where the graph crosses the y-axis: the point (0, b).

x-intercept

Where the graph crosses the x-axis: set y = 0 and solve.

Rise over run

Slope as a staircase: rise units up for every run units across.

Direct substitution

Plugging a point (x, y) into y = mx + b to solve for the missing piece.

Practice slope-intercept Open the line grapher Explore in the lab