A line in the coordinate plane has one shape but three common equation forms — and each is the right choice for a specific question. Knowing which to reach for first will save time on every algebra and geometry problem you ever solve.
The three forms
- Slope-intercept: y = mx + b
- Point-slope: y − y₁ = m(x − x₁)
- Standard form: Ax + By = C (with A, B, C integers and A ≥ 0)
All three describe the same lines. The differences are in what's exposed in the equation.
Slope-intercept: best for graphing
y = mx + b shows the slope (m) and y-intercept (b) directly. This makes it the easiest form for sketching a graph: plot b on the y-axis, then use m as rise/run to find a second point.
Example: y = 2x + 3 has slope 2 (rise 2, run 1) and y-intercept 3. Plot (0, 3), then go up 2 right 1 to (1, 5). Connect.
When to use: graphing a line, identifying y-intercept, comparing slopes of multiple lines side by side.
Point-slope: best for writing an equation given a point
Point-slope form, y − y₁ = m(x − x₁), takes a slope m and any point (x₁, y₁) on the line. It's the natural form when you know one point and the slope, but not the y-intercept.
Example: line passes through (3, 7) with slope 4.
- Point-slope: y − 7 = 4(x − 3)
- Convert to slope-intercept: y = 4x − 12 + 7 = 4x − 5
When to use: answering "write the equation of a line through (a, b) with slope m" — point-slope is the form your teacher wants you to write down first.
Standard form: best for finding intercepts and integer arithmetic
Ax + By = C looks ugly but it's the form most teachers want answers simplified to on tests. It also makes intercepts easy:
- x-intercept: set y = 0, solve Ax = C → x = C/A
- y-intercept: set x = 0, solve By = C → y = C/B
Example: 2x + 3y = 12.
- x-intercept: x = 12/2 = 6, so (6, 0)
- y-intercept: y = 12/3 = 4, so (0, 4)
When to use: finding both intercepts quickly, fitting a "clean" textbook form, comparing two lines' coefficients.
Converting between forms
Point-slope → slope-intercept: distribute and solve for y. y − 7 = 4(x − 3) → y − 7 = 4x − 12 → y = 4x − 5.
Slope-intercept → standard form: move x to the left. y = 4x − 5 → −4x + y = −5 → 4x − y = 5 (multiply by −1 to make A positive).
Standard form → slope-intercept: solve for y. 4x − y = 5 → −y = −4x + 5 → y = 4x − 5.
The slope from standard form: m = −A / B. The y-intercept: b = C / B. Memorize these.
Worked example: from two points to all three forms
Line through (1, 2) and (4, 8).
- Slope: m = (8 − 2) / (4 − 1) = 6 / 3 = 2.
- Point-slope: using (1, 2), y − 2 = 2(x − 1).
- Slope-intercept: y − 2 = 2x − 2 → y = 2x. So slope 2, y-intercept 0.
- Standard form: 2x − y = 0 (or −2x + y = 0; convention is A ≥ 0, so 2x − y = 0).
Special cases
Horizontal line (slope 0): y = b. In standard form: 0x + y = b.
Vertical line (undefined slope): x = a. Cannot be written in slope-intercept form. In standard form: x + 0y = a.
SAT geometry problems often involve a vertical line just to test whether you remember it has no slope-intercept form.
Common mistakes
- Forgetting to convert standard form to slope-intercept when you need the slope. m = −A/B, not just A.
- Mixing up the signs in point-slope: y − y₁ = m(x − x₁), with subtraction. y₁ = −3 means y + 3 in the formula.
- Leaving fractions in standard form. 2x − (3/4)y = 1 should be cleared: 8x − 3y = 4.
Compute the slope first
Whichever form you need, the slope is your first step. Our slope calculator takes any two points and returns slope, y-intercept, and the slope-intercept equation in one shot. Useful for verifying homework answers or generating practice equations from custom point pairs.