Calculator

Slope Calculator with Steps

Enter two points and get the slope, midpoint, and distance, then the line itself in point-slope, slope-intercept, and standard form — plus the parallel and perpendicular slopes.

Solve a coordinate geometry problem now

First solution free

How to use it

Give the two points as ordered pairs, keeping the signs attached. If you have one point and a slope instead, say so and the working starts from point-slope form. Ask for the perpendicular or parallel line through a given point in the same question, since it reuses the slope you already computed. Coordinate proofs involving midpoints and distances are handled by the geometry solver.

The method behind it

Slope is the constant ratio of vertical change to horizontal change along a line. Because the ratio is constant, any two points on the line produce the same value, which is what makes the formula well defined.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = \frac{rise}{run}

The sign tells you the direction and the size tells you the steepness. A slope of zero is a horizontal line, since the rise is zero. A vertical line has zero run, and division by zero is undefined — that is a genuinely different situation from a slope of zero, not a larger version of it.

Once you have the slope and any one point, the line is determined. Point-slope form is the honest statement of that fact, and it is the form to write first because it needs no rearrangement.

y - y_{1} = m (x - x_{1})

Expanding gives slope-intercept form $y = m x + b$ , which is what you want for graphing. Clearing fractions and moving the variables to one side gives standard form $A x + B y = C$ , which is what you want for systems and for lines that happen to be vertical.

The same two points also give the midpoint, by averaging each coordinate, and the distance, by applying Pythagoras to the horizontal and vertical gaps.

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}), d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Two lines are parallel when their slopes are equal and perpendicular when the product of their slopes is −1, which is the same as saying one slope is the negative reciprocal of the other. The negative and the reciprocal are two separate operations and both are required.

parallel: m_{1} = m_{2} perpendicular: m_{1} m_{2} = - 1

This ratio is also the first appearance of an idea that runs through the rest of mathematics: it is the average rate of change of a linear function, and the derivative that calculus builds is exactly this quantity for curves that do not stay straight. Where two lines meet is a system of equations, and rate-of-change problems phrased as stories are usually a translation exercise before they are an arithmetic one.

Worked examples

Worked example

A tank holds 480 L at 09:00 and 300 L at 12:00. Find the rate and the emptying time.

1
Turn the times into coordinates by measuring hours from the first reading. Volume is the dependent variable, so it goes second.
$(0, 480) and (3, 300)$
2
Apply the slope formula. Keeping the subtraction in the same order in both the numerator and the denominator is what gets the sign right.
$m = \frac{300 - 480}{3 - 0} = \frac{- 180}{3} = - 60$
3
Read the units off the axes: litres in the rise, hours in the run. The tank loses 60 litres every hour, and the negative sign is the loss.
$m = - 60 L per hour$
4
The vertical intercept is the volume at time zero, which was given, so slope-intercept form can be written immediately.
$V = 480 - 60 t$
5
Empty means zero volume. Solve for t and check the units: 480 litres divided by 60 litres per hour leaves hours, as it must.
$0 = 480 - 60 t \Rightarrow t = 8 hours after 09:00, i.e. 17:00$

Answer

m = - 60 L/h, V = 480 - 60 t, empty at t = 8

Worked example

Find the line perpendicular to 2 x + 5 y = 20 through (4, - 1) .

1
Standard form hides the slope, so rearrange into slope-intercept form first.
$5 y = - 2 x + 20 \Rightarrow y = - \frac{2}{5} x + 4$
2
The given slope is −2/5. Take the negative reciprocal: flip the fraction and change the sign.
$m_{⊥} = - \frac{1}{- \frac{2}{5}} = \frac{5}{2}$
3
Use point-slope form with the new slope and the given point. Subtracting a negative y-coordinate makes it an addition.
$y - (- 1) = \frac{5}{2} (x - 4)$
4
Expand and simplify to slope-intercept form.
$y + 1 = \frac{5}{2} x - 10 \Rightarrow y = \frac{5}{2} x - 11$
5
Multiply by 2 for standard form, and verify: the slopes multiply to −1, and the point satisfies the equation.
$5 x - 2 y = 22, (- \frac{2}{5}) (\frac{5}{2}) = - 1$

Answer

y = \frac{5}{2} x - 11, 5 x - 2 y = 22

Everything from two points

One pair of coordinates answers five separate exam questions. Work through them in this order and each result feeds the next, which is faster than treating them as unrelated formulas.

From the points (−3, 7) and (5, −1)

0 of 7 steps

Try it yourself first — you'll remember it longer.

Common mistakes

Subtracting in inconsistent directions

$\frac{y _{2} - y _{1}}{x _{1} - x _{2}}$ produces the slope with the wrong sign. Both differences must run from the same point to the same point. Writing the two coordinates one above the other and subtracting column by column makes the direction impossible to lose.

Treating a vertical line as having slope zero

A horizontal line has slope 0 and equation $y = k$ . A vertical line has undefined slope and equation $x = k$ , and it has no slope-intercept form at all. The two cases are opposites, not variations.

Half-completing the negative reciprocal

The perpendicular slope to $- \frac{3}{7}$ is $\frac{7}{3}$ . Flipping without changing the sign gives $- \frac{7}{3}$ , and negating without flipping gives $\frac{3}{7}$ . Multiply your answer by the original slope; it must come to exactly −1.

Practice

Answers are checked here — nothing is sent anywhere.

1
$Slope through (- 1, 4) and (3, - 8)$
2
$Midpoint of (- 5, 2) and (9, - 6)$
3
$Slope of a line perpendicular to y = - \frac{3}{7} x + 1$

Frequently asked questions

What happens when the two points have the same x-coordinate?

The run is zero and the slope is undefined, not infinite and not zero. The line is vertical and its equation is x equals that shared value, which cannot be written in slope-intercept form at all.

Does it give the line in every form?

Point-slope, slope-intercept, and standard form all come from the same calculation, so all three are shown. Standard form is written with integer coefficients and a positive leading coefficient, which is the convention most courses mark against.

Will it also find the midpoint and the distance?

Yes, from the same pair of points. Midpoint averages the coordinates; distance applies Pythagoras to the horizontal and vertical gaps. Ask for all three at once rather than in three separate questions.

Is slope the same thing as rate of change?

Yes, and the units make that concrete. For a graph of litres against hours, the slope is litres per hour. Attaching units to the rise and run is the fastest way to check a word problem answer is plausible.

What does it cost?

The first solution is free and needs no account. A free account gives three solutions a day, and Gauth Plus is unlimited at $11.99 a month with a three-day free trial.

Keep going

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