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Point-Slope Form Calculator

Enter a point and a slope to build the line's equations.

Result

y = 1.5000x − 6.5000
b = -6.5000
y + 2.0000 = 1.5000(x − 3.0000)

How it is calculated

A line through the point (x₁, y₁) with slope m satisfies y − y₁ = m(x − x₁). Expanding the bracket and moving y₁ to the right gives y = mx + b with b = y₁ − m·x₁, the value of y where the line crosses the y-axis. The tool computes b this way and prints three things: the slope-intercept form, the intercept b, and the point-slope form.

Numbers are shown with four decimals. Signs are written out so that a negative point or intercept appears as + or − between the terms instead of a double sign. All three fields must be filled in. Vertical lines have no finite slope, so they cannot be entered.

Worked example

Take the point (3, −2) with slope 1.5. First b = y₁ − m·x₁ = −2 − 1.5 × 3 = −2 − 4.5 = −6.5. So the line is y = 1.5000x − 6.5000. The point-slope form is y − (−2) = 1.5(x − 3), which the tool prints as y + 2.0000 = 1.5000(x − 3.0000). A check: at x = 3, y = 1.5 × 3 − 6.5 = −2, so the line does pass through the point.

Try this example →

Common mistakes

  • Losing the sign of the point. With y₁ = −2 the form is y + 2 = …, not y − 2 = …; likewise a negative x₁ turns the bracket into (x + …).
  • Entering the slope as a rounded decimal when it is a fraction such as 1/3. Typing 0.33 shifts b noticeably if x₁ is large; enter more digits, because the output shows only four decimals.
  • Swapping the coordinates, entering (y, x) instead of (x, y). The slope then applies to the wrong axis and b is wrong.
  • Trying to describe a vertical line (such as x = 5). Its slope is undefined, so it has no point-slope or slope-intercept form.
  • Mixing up b with the point's y-value. b is the y-intercept at x = 0, which only equals y₁ when x₁ = 0.

About this calculator

The point-slope form of a line, y − y1 = m(x − x1), is the most direct way to write a linear equation when you know one point on the line and its slope — no need to first find the y-intercept. Rearranging it algebraically gives the more familiar slope-intercept form, y = mx + b, which is easier to read values off of and to graph.

This calculator takes a point and a slope and produces both equations, showing the algebra step in between. It’s a staple of algebra and pre-calculus coursework, and it comes up constantly outside the classroom too — in physics and engineering whenever a linear relationship is defined by a single measured point and a rate of change, such as fitting a trend line to two data readings.

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Comments (1)

  • Editorial team

    Verify the line by stepping one unit right of your point: y should rise by exactly the slope. With the defaults, (2, 3) and slope 4, the point x = 3 must give y = 7, and y = 4x − 5 does give 7. Enter fractions such as 1/3 as long decimals, since the output shows four. If you start from two points rather than a slope, tell us whether you want that mode.

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