Point-Slope Form Calculator

What point-slope form is

Point-slope form is one of three standard ways to write the equation of a straight line. It says y - y1 = m(x - x1), where m is the slope and (x1, y1) is any single point the line passes through. If you know where the line goes and how steep it is, this form hands you the equation with almost no algebra at all.

The form comes straight from the definition of slope. Slope is rise over run, m = (y - y1) / (x - x1), because it compares the vertical change to the horizontal change between two points on the line. Multiply both sides by (x - x1) and the slope definition turns into point-slope form. That is the entire derivation.

The calculator below returns point-slope, slope-intercept and standard form at once, along with the y-intercept on its own. Its conventions match the lines chapter of Paul's Online Notes at Lamar University, which works through the same algebra in more depth.

How to use the calculator

The widget has two modes. In the point and slope mode you type the slope m and one point (x1, y1). In the two points mode you type both points and the calculator derives the slope for you, then writes the equation from whichever point you listed first. Results update as you type, and blank inputs count as 0.

You get four rows back: the point-slope form, the slope-intercept form y = mx + b, the standard form ax + by = c, and the y-intercept b on its own. The point-slope row is built from your inputs with no rearranging, so it is the fastest one to copy into homework that asks for point-slope form specifically.

One habit worth building: check the answer by feeding the x coordinate of your point back into the slope-intercept form. If the point is (3, 4) and the output is y = 2x - 2, then at x = 3 the formula gives y = 2 x 3 - 2 = 4, which lands on your point. If it does not, a sign got flipped somewhere.

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Worked example: point (3, 4) with slope 2

Take the point (3, 4) and slope 2. Substituting into y - y1 = m(x - x1) gives y - 4 = 2(x - 3), and that is already a complete answer for a question that asks for point-slope form. Most textbooks, though, want it simplified into slope-intercept form.

Two checks confirm the line. The y-intercept should be b = -2, and plugging x = 0 into y = 2x - 2 gives -2. The original point should satisfy the equation: 2 x 3 - 2 = 4, which matches y1. A third point keeps it honest: at x = 5 the line gives y = 2 x 5 - 2 = 8, so (5, 8) sits on it too.

The figure below draws the line. The red dot is the point (3, 4) you started from, and the green dashed legs show the rise of 4 over a run of 2 that the slope describes.

From two points to point-slope form

When a problem gives two points instead of a slope, compute the slope first, then treat one of the points as (x1, y1). Take (1, 2) and (4, 11). The slope is m = (11 - 2) / (4 - 1) = 9 / 3 = 3.

Test the result against the point you did not use. At x = 4 the formula gives y = 3 x 4 - 1 = 11, which matches the second point. That check matters because the most common two-point error is dividing in the wrong order; if you had written (4 - 1) / (11 - 2) you would get a slope of 1/3 and the check would fail.

Either point produces a correct point-slope form. Starting from (4, 11) instead gives y - 11 = 3(x - 4), which expands to the same y = 3x - 1. The calculator uses your first point and shows the equivalent version built from the second one.

Point-slope, slope-intercept and standard form compared

All three forms describe the same object. They differ in what they make easy to read off, and teachers ask for specific ones because each shines in a different situation.

Using the earlier example, one line has three costumes: y - 4 = 2(x - 3), y = 2x - 2 and 2x - y = 2. Graphing calculators usually want slope-intercept or the solved form, while elimination in a system moves fastest in standard form. Point-slope is the natural first draft, because it is the only one you can write down the instant you know a point and a slope.

The linear equations and graphs unit on Khan Academy covers the same three forms with extra practice sets, and our standard form to slope-intercept converter handles the conversion in the other direction.

Parallel and perpendicular lines in point-slope form

Parallel lines share the same slope, so a parallel line is a copy of the original equation with a different point swapped in. A line parallel to y = 2x - 2 through the point (1, 5) is y - 5 = 2(x - 1), which simplifies to y = 2x + 3. Same m, different line.

Perpendicular lines meet at a right angle, and their slopes multiply to -1. So the perpendicular slope is the negative reciprocal: flip the fraction and change the sign. The perpendicular partner of slope 2 is -1/2, because 2 x -0.5 = -1.

Concretely: find the line through (5, 1) perpendicular to y = 2x - 2. The slope is -0.5, so point-slope form gives y - 1 = -0.5(x - 5). Distribute: y - 1 = -0.5x + 2.5. Add 1: y = -0.5x + 3.5. A quick check: at x = 5 the formula gives -0.5 x 5 + 3.5 = 1, which lands on the point.

Horizontal and vertical lines are the two special cases. A horizontal line has slope 0, so point-slope form collapses to y = y1, for example y = 4. A vertical line has an undefined slope and cannot be written in point-slope or slope-intercept form at all; its equation is just x = x1, for example x = 3.

Common mistakes and how to catch them

Point-slope form is short, which leaves room for a small set of reliable errors. The subtraction on both sides is where most of them live, because subtracting a negative coordinate looks like an addition mistake even when it is correct.

The single fastest check is substitution. Pick any point you trust, the one you started with or the second of a pair, put its x into your simplified equation, and confirm the y comes back out. It takes ten seconds and catches every one of the mistakes in the table above.

For a different angle on the same idea, the slope calculator on this site starts from two points and returns the slope, angle, distance and grade together, which is handy when the problem is about steepness rather than the equation.