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Equation of a straight line calculator

Choose what you know about the line (two points, a point and its slope, its intercepts, or a line it must be parallel or perpendicular to) and get its equation in all four standard forms, with exact fractions and the working shown.

Equation of the line y = −2x + 3
General form2x + y − 3 = 0
Slope−2
x-intercept3/2
y-intercept3
FormEquation
Slope-intercept (y = mx + c)y = −2x + 3
Point-slope (y − y₁ = m(x − x₁))y − 5 = −2(x + 1)
General (ax + by + c = 0)2x + y − 3 = 0
Intercept (x/a + y/b = 1)x/(3/2) + y/3 = 1
-8-6-4-20246810-4-20246 (−1, 5)(3, −3)x-int 3/2y-int 3 x y
Show the working, step by step

  1. Slope from the two points: m = (y₂ − y₁) ÷ (x₂ − x₁).

    m = (−3 − 5) ÷ (3 − (−1)) = −8 ÷ 4 = −2

  2. Rearrange into the other forms.

    slope-intercept: y = −2x + 3 general: 2x + y − 3 = 0 intercept: x/(3/2) + y/3 = 1

The four forms

Slope-intercept: y = mx + c Point-slope: y − y₁ = m(x − x₁) General: ax + by + c = 0 Intercept: x/a + y/b = 1

All four describe the same line. Slope-intercept shows the gradient and where the line crosses the y-axis. Point-slope is the quickest to write down from a point and a slope. General form is what distance and intersection formulas expect, and it handles vertical lines. Intercept form shows both axis crossings at once.

A worked example

The calculator opens with the points (−1, 5) and (3, −3).

  1. Slope: m = (−3 − 5) ÷ (3 − (−1)) = −8 ÷ 4 = −2.
  2. Point-slope with (−1, 5): y − 5 = −2(x + 1).
  3. Expand: y = −2x − 2 + 5, so y = −2x + 3.
  4. General form: 2x + y − 3 = 0.
  5. Intercepts: y = 0 gives x = 3/2, and x = 0 gives y = 3, so x/(3/2) + y/3 = 1.

The other modes open on the same line so you can compare them. Slope −2 through (−1, 5), slope −2 with y-intercept 3, and intercepts 3/2 and 3 all give y = −2x + 3. So does the line through (−1, 5) perpendicular to x − 2y + 4 = 0, because that line has slope 1/2 and (1/2) × (−2) = −1.

Which method to use

You knowStart from
Two pointsm = (y₂ − y₁)/(x₂ − x₁), then point-slope
A point and the slopey − y₁ = m(x − x₁)
The slope and the y-intercepty = mx + c directly
Both interceptsx/a + y/b = 1
A point and a parallel lineSame a and b as the given line, new c
A point and a perpendicular lineSwap a and b, change one sign, new c

Common mistakes

  • Sign slips in point-slope form. For the point (−1, 5) the bracket is (x + 1), not (x − 1).
  • Leaving fractions in the general form. Multiply through so a, b and c are whole numbers with no common factor.
  • Using the negative slope for a perpendicular line. The perpendicular slope is the negative reciprocal, −1/m. For m = 1/2 that is −2, not −1/2.
  • Reading c in ax + by + c = 0 as the y-intercept. The y-intercept is −c/b. For 2x + y − 3 = 0 it is 3.

Common questions

How do you find the equation of a line through two points?

Find the slope m = (y₂ − y₁)/(x₂ − x₁), then substitute one point into y − y₁ = m(x − x₁) and rearrange. Through (−1, 5) and (3, −3): m = −8/4 = −2, so y − 5 = −2(x + 1), which gives y = −2x + 3, or 2x + y − 3 = 0.

What is the general form of a straight line?

ax + by + c = 0, usually written with whole-number coefficients, no common factor and a positive leading coefficient. Unlike y = mx + c it can describe vertical lines (b = 0). Clear fractions by multiplying through by the lowest common denominator: y = (2/3)x − 7/3 becomes 2x − 3y − 7 = 0.

What is the intercept form of a line?

x/a + y/b = 1, where a is the x-intercept and b is the y-intercept. The line y = −2x + 3 crosses the axes at (3/2, 0) and (0, 3), so its intercept form is x/(3/2) + y/3 = 1. Lines through the origin have no intercept form, because both intercepts are 0.

How do I find a line parallel or perpendicular to a given line?

A parallel line keeps the same x and y coefficients: parallel to x − 2y + 4 = 0 means x − 2y + k = 0. A perpendicular line swaps them and changes one sign: 2x + y + k = 0. Then put the given point in to find k. Through (−1, 5): the parallel line is x − 2y + 11 = 0 and the perpendicular line is 2x + y − 3 = 0.

What is the slope of a vertical line?

It is undefined, because the run x₂ − x₁ is zero. A vertical line has the equation x = constant and cannot be written as y = mx + c. A horizontal line has slope 0 and equation y = constant.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.