Algebra 1 · Linear equations

Slope-Intercept & Point-Slope

Two ways to write the same straight line. Learn what each piece means, play with the grapher, then practice until the conversions feel automatic.

The two forms

Every non-vertical line has one slope, m. The two forms just describe the line from different starting spots.

Slope-intercept

y = mx + b

m = slope = rise ÷ run
b = y-intercept, where the line crosses the y-axis, the point (0, b)

Use it when you know the slope and where the line crosses the y-axis. Best form for graphing fast.

Point-slope

y − y₁ = m(x − x₁)

m = slope
(x₁, y₁) = any one point on the line

Use it when you know a point and the slope, or two points. Then simplify to slope-intercept.

Slope from two points: m = y₂ − y₁x₂ − x₁ Same order on top and bottom.

Grapher

Drag the points or use the sliders. Blue moves the y-intercept, green slides along the line as (x₁, y₁), and gold sets the rise and run. Watch both equations update.

Rise

Run

y-intercept b

Point x₁

Slope

Slope-intercept

Point-slope through (x₁, y₁)

Point-slope → slope-intercept

Example: slope 3, through the point (2, −4).

y − (−4) = 3(x − 2)Plug m and the point into point-slope.
y + 4 = 3(x − 2)Minus a negative is plus.
y + 4 = 3x − 6Distribute the 3 to both terms inside.
y = 3x − 10Subtract 4 from both sides. Done: m = 3, b = −10.
−4 ≟ 3(2) − 10 = −4 ✓Check by plugging the original point back in.

Shortcut for b: b = y₁ − m·x₁ → −4 − 3(2) = −10.

Common traps

Sign flip on the pointy + 3 = 2(x − 5) means the point is (5, −3), not (5, 3). The form has minus signs built in.

Forgetting to distribute2(x − 5) is 2x − 10, not 2x − 5.

Slope upside downSlope is change in y over change in x. Rise on top.

Mixed subtraction orderIf you do y₂ − y₁ on top, you must do x₂ − x₁ on the bottom.

Using the point's y as bb is the y-value only where x = 0. A random point's y-value is not the intercept.

Practice

Find the slope m and y-intercept b. Type fractions like -3/2.

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