Slope Calculator
Enter two points to find the slope of the line through them, plus its equation in slope-intercept, point-slope and standard form. Every value stays an exact fraction and every step is shown.
Each box takes an integer (−3), a decimal (2.5) or a fraction (3/4).
Negative slope, falls left to right
Step by step
- Slope formula: m = (y₂ − y₁)/(x₂ − x₁).
- Substitute the points (1, 2) and (5, −4): m = (−4 − 2)/(5 − 1).
- Rise and run: rise = y₂ − y₁ = −6, run = x₂ − x₁ = 4.
- Simplify: m = −6/4 = −3/2.
- Point-slope form with point 1, y − y₁ = m(x − x₁): y − 2 = −(3/2)(x − 1).
- Solve for y. The y-intercept is b = y₁ − m·x₁ = 2 − (−3/2)·1 = 7/2, so y = −(3/2)x + 7/2.
- Standard form Ax + By = C: multiply by the denominators to clear fractions, move the x term to the left and divide out any common factor, keeping A positive: 3x + 2y = 7.
- Intercepts: set x = 0 to get the y-intercept (0, 7/2), and set y = 0 to get the x-intercept (7/3, 0).
How it works
The slope of a line is rise over run: the change in y divided by the change in x between two points, m = (y₂ − y₁)/(x₂ − x₁). Once you know the slope and one point, the rest of the line follows. Point-slope form, y − y₁ = m(x − x₁), drops straight out of the formula. Solving it for y gives slope-intercept form, y = mx + b, and clearing fractions gives standard form, Ax + By = C.
The calculator does all of this with exact fractions, so 1/3 stays 1/3 instead of turning into 0.333333. Two special cases get their own wording. A vertical line has a run of 0, so its slope is undefined and its equation is x = c. A horizontal line has a rise of 0, so its slope is 0 and its equation is y = c.
Worked example
Find the line through (1, 2) and (5, −4).
- Rise: y₂ − y₁ = −4 − 2 = −6. Run: x₂ − x₁ = 5 − 1 = 4.
- Slope: m = −6/4 = −3/2. The line falls 3 units for every 2 units to the right.
- Point-slope form with (1, 2): y − 2 = −(3/2)(x − 1).
- Solve for y: b = 2 − (−3/2)(1) = 2 + 3/2 = 7/2, so the slope-intercept form is y = −(3/2)x + 7/2.
- Standard form: multiply by 2 to get 2y = −3x + 7, then add 3x to both sides: 3x + 2y = 7.
- Intercepts: setting x = 0 gives the y-intercept 7/2. Setting y = 0 gives 3x = 7, so the x-intercept is 7/3.
Those are the default values above, so you can compare each line of the working with the calculator's steps. For more practice reading slope and b off an equation, see the guide to slope and slope-intercept form. The point-slope form guide covers writing a line from one point and a slope, and the guide to graphing linear equations shows how to sketch the line by hand. More calculators are on the free tools page.
Frequently asked questions
What does the slope of a line mean?
Slope measures how steep a line is and which way it tilts. It is the change in y divided by the change in x between any two points on the line, often called rise over run. A slope of −3/2 means that every time x goes up by 2, y goes down by 3. Positive slopes rise from left to right and negative slopes fall.
What is the difference between an undefined slope and a zero slope?
A zero slope belongs to a horizontal line such as y = 4: the rise is 0, so 0 divided by any run is 0. An undefined slope belongs to a vertical line such as x = 3: the run is 0, and dividing by zero has no answer. A vertical line has no slope-intercept form, so its equation is written as x = c.
Does the order of the two points matter?
No. If you swap the points, the rise and the run both change sign, so their ratio stays the same. With (1, 2) and (5, −4) you get −6/4, and with the points swapped you get 6/(−4), which is the same −3/2. Just keep the order consistent: subtract the same point first in both the top and the bottom.
Why does the calculator show fractions instead of decimals?
Many slopes, like 1/3 or 7/6, have decimals that never end, so a rounded decimal is not quite the right line. The calculator keeps every value as an exact fraction, which is also the form most teachers expect in an equation like y = −(3/2)x + 7/2. The decimal is shown next to the slope when it helps you picture it.
How do I convert an equation to standard form?
Standard form is Ax + By = C with whole numbers. Start from slope-intercept form, multiply every term by the lowest common denominator to clear fractions, then move the x term to the left side. For y = −(3/2)x + 7/2, multiplying by 2 gives 2y = −3x + 7, and adding 3x to both sides gives 3x + 2y = 7.
What is the angle of inclination?
It is the angle the line makes with the positive x-axis, measured counterclockwise, from 0° up to but not including 180°. Its tangent equals the slope, so a slope of 1 gives 45° and a slope of −3/2 gives about 123.69°. A horizontal line has an angle of 0° and a vertical line has an angle of 90°.
Check your algebra homework
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