Point-Slope Form Calculator
Enter a slope and a point, or two points, to get the line in every standard form: point-slope, slope-intercept, and standard form. The calculator also finds the x-intercept and y-intercept, the angle of inclination, and the slopes of parallel and perpendicular lines.
Formula
Worked example
Slope m = 2 through (3, 1): point-slope is y - 1 = 2(x - 3). Slope-intercept: y = 2x - 5. Standard form: -2x + y = -5 (or 2x - y = 5). The x-intercept is 5/2 = 2.5, the angle is arctan(2) = 63.43 degrees, and a perpendicular line has slope -0.5.
What point-slope form is and why it matters
Point-slope form, written y - y1 = m(x - x1), is the most direct way to write a line when you know its slope and one point on it. The form comes straight from the definition of slope: rearranging m = (y - y1) / (x - x1) by multiplying both sides by (x - x1) gives the point-slope equation. It appears in algebra, calculus (tangent lines), and statistics (regression lines through the mean). Every non-vertical line can be written this way, and converting to other standard forms is straightforward once you have it.
The four line forms and when to use each
Point-slope form y - y1 = m(x - x1) is best when you start from a known point and slope. Slope-intercept form y = mx + b is the easiest to graph because you can read off the y-intercept b immediately and count the rise/run from there. Standard form Ax + By = C with integer coefficients is common in textbooks and systems of equations because both variables are on the same side, making matrix methods tidy. General form Ax + By + C = 0 is a common variant where everything is on one side. All four forms describe the same unique line; the choice is about what information you already have and what you need next.
Intercepts, angle, and perpendicular slope
The x-intercept is where the line crosses the x-axis, found by setting y = 0 in y = mx + b, giving x = -b/m. The angle of inclination is the angle the line makes with the positive x-axis, computed as arctan(m) in degrees. A perpendicular line crosses at a right angle; its slope is the negative reciprocal -1/m. Parallel lines have the same slope m. These values come up constantly in geometry, physics (projectile angle, gradient), and engineering (grade of a road or pipe).
Using two-point mode
If you know two points (x1, y1) and (x2, y2) but not the slope, switch to "Two points" mode. The calculator first computes the slope as m = (y2 - y1) / (x2 - x1), then uses the first point to build the point-slope equation. The fraction form of the slope is shown in the steps panel so you can verify it matches what you expect. If x1 equals x2 the line is vertical and the slope is undefined; the calculator returns a blank in that case.
Line equation forms at a glance
| Form | Equation | Best used when |
|---|---|---|
| Point-slope | y - y1 = m(x - x1) | You know slope m and one point |
| Slope-intercept | y = mx + b | You want to graph the line quickly |
| Standard form | Ax + By = C | Working with systems of equations |
| General form | Ax + By + C = 0 | Textbook problems, matrix methods |
All four forms describe the same line; the choice depends on what data you have.
Frequently asked questions
What is the point-slope formula?
The point-slope form of a line is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a known point on the line. Substitute your slope and point to get the equation instantly.
How do I convert point-slope to slope-intercept form?
Distribute the slope across the parentheses, then add y1 to both sides: y = mx + (y1 - m*x1). The y-intercept is b = y1 - m*x1. This calculator shows that step automatically.
How do I convert point-slope to standard form?
Start from y = mx + b, then move the x-term to the left: -mx + y = b. Multiply every term by -1 (or by a common factor to clear fractions), giving Ax + By = C with integer coefficients. The calculator does this for you and shows the steps.
What is the x-intercept and how is it found?
The x-intercept is the point where the line crosses the x-axis (where y = 0). Set y = 0 in y = mx + b and solve: x = -b/m. This calculator reports both intercepts automatically.
What is the angle of inclination?
The angle of inclination is the angle the line makes with the positive x-axis, measured in degrees. It equals arctan(m). A slope of 1 gives 45 degrees; a slope of 0 gives 0 degrees (horizontal). Negative slopes give angles between -90 and 0 degrees.
How do I find a perpendicular slope?
A line perpendicular to a line with slope m has slope -1/m (the negative reciprocal). If the original slope is 2, the perpendicular slope is -1/2. Perpendicular lines intersect at exactly 90 degrees.
Can point-slope form describe a vertical line?
No. A vertical line has an undefined slope (its run is zero), so m does not exist and the form cannot be used. Vertical lines are written as x = constant, such as x = 4.