Point-Slope Form Calculator
Build a line equation from Point + Slope or Two Points. Get point-slope, slope-intercept, and standard form, the y-intercept and x-intercept, a to-scale mini graph, and a step-by-step solution that shows the work.
Background
Point-slope form is y − y₁ = m(x − x₁). It’s the fastest way to write a line when you know a point on the line and the slope. To read the y-intercept, rearrange it into slope-intercept form, y = mx + b, where b is the y-intercept.
How to use this calculator
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Choose what you know.
Select Point + slope if you know a point and the slope, Two points if you know two points, or Undefined slope for a vertical line like x = 4. -
Enter the values.
Type whole numbers, decimals like -1.5, or fractions like -3/4. The calculator keeps results exact when possible. -
Adjust options (optional).
Choose whether to round decimals, show step-by-step work, or display the mini graph. -
Click Calculate.
Instantly get the line written in point-slope, slope-intercept, and standard form, along with the y-intercept and x-intercept. If you change a number later, click Calculate again to update the result. -
Explore the results.
Follow the step-by-step solution to see how each form and intercept was found, hover over the graph to see the equation, and use the extra tools to check a point or build parallel and perpendicular lines.
Tip: Try the quick-pick examples to see common cases like vertical lines, horizontal lines, or fractional slopes.
How this calculator works
- Point + slope: plugs into y − y₁ = m(x − x₁).
- Two points: slope is m = (y₂ − y₁)/(x₂ − x₁), then we use point-slope.
- Slope-intercept: compute b = y₁ − mx₁ → y = mx + b.
- Intercepts: the y-intercept is b, and the x-intercept is −b/m (set y = 0).
- Standard form: rearrange into Ax + By = C with integer coefficients.
Tip: A vertical line has undefined slope and looks like x = 4. A horizontal line has slope 0 and looks like y = 3.
Formula & Equation Used
Point-slope: y − y₁ = m(x − x₁)
Slope from two points: m = (y₂ − y₁)/(x₂ − x₁)
Slope-intercept: b = y₁ − mx₁ and y = mx + b
Y-intercept: b, the point (0, b)
X-intercept: x = −b/m, the point (−b/m, 0)
Standard form: Ax + By = C
Example Problem & Step-by-Step Solution
Example 1 — Two points (1, 2) and (5, −4)
- Compute slope: m = (−4 − 2)/(5 − 1) = −6/4 = −3/2
- Point-slope (use point (1,2)): y − 2 = (−3/2)(x − 1)
- Slope-intercept: b = 2 − (−3/2)·1 = 7/2 → y = (−3/2)x + 7/2
- Intercepts: the y-intercept is (0, 7/2). For the x-intercept, set y = 0: 0 = (−3/2)x + 7/2, so x = 7/3, the point (7/3, 0).
- Standard form: 3x + 2y = 7
Example 2 — Point + slope (2, −1) with m = 3
- Start with point-slope form: y − y₁ = m(x − x₁)
- Substitute: y − (−1) = 3(x − 2) → y + 1 = 3(x − 2)
- Distribute: y + 1 = 3x − 6
- Solve for y: y = 3x − 7, so the y-intercept is (0, −7)
- X-intercept (set y = 0): 0 = 3x − 7 → x = 7/3
- Standard form: 3x − y = 7
Example 3 — Vertical line through (4, 1) and (4, 7)
- Compute slope: m = (y₂ − y₁)/(x₂ − x₁)
- Here x₂ − x₁ = 4 − 4 = 0, so the slope is undefined.
- A line with undefined slope is vertical.
- Equation: x = 4. The x-intercept is (4, 0), and there is no y-intercept because x is never 0.
Example 4 — Horizontal line through (−1, 3) with m = 0
- Substitute: y − 3 = 0(x + 1)
- Anything times 0 is 0, so y − 3 = 0 → y = 3
- The y-intercept is (0, 3). A flat line never crosses the x-axis, so there is no x-intercept.
Frequently Asked Questions
Q: What is point-slope form?
Point-slope form is y − y₁ = m(x − x₁). It writes the equation of a line using the slope m and one point (x₁, y₁) on the line. You can rearrange it into slope-intercept form, y = mx + b, or standard form, Ax + By = C.
Q: How do I find the y-intercept?
The y-intercept is where the line crosses the y-axis, at x = 0. Rearrange to y = mx + b and read b, or compute b = y₁ − mx₁. The calculator shows it as a labelled result and marks it on the graph.
Q: How do I find the x-intercept?
Set y = 0 and solve for x, which gives x = −b/m. A horizontal line (slope 0) never crosses the x-axis unless it is the x-axis itself, and a vertical line x = c crosses it at (c, 0).
Q: Why does my slope show “undefined”?
If x₁ = x₂, the denominator in (y₂ − y₁)/(x₂ − x₁) is zero, so the slope is undefined and the line is vertical.
Q: Why do you “prefer exact fractions”?
Fractions keep answers clean (like −3/2) and avoid rounding errors. You can still show decimals with rounding.
Q: Can I use this calculator to find parallel or perpendicular lines?
Yes. After calculating the original line, you can generate a parallel line (same slope) or a perpendicular line (negative reciprocal slope) through any new point. The calculator outputs the equations and shows them on the mini graph.
Q: What happens if the line is vertical or horizontal?
If a line is vertical, its equation is x = c and the slope is undefined. If a line is horizontal, its equation is y = c and the slope is 0. The calculator automatically detects both cases and displays the correct equation, intercepts, and graph.
Q: Why did my result turn grey?
You changed one of the numbers after calculating, so the result on screen no longer matches the inputs. Click Calculate to update it.
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