The Slope Formula

Slope measures steepness: how much a line goes up (or down) for every step it goes across.

m = y_2 - y_1x_2 - x_1

Slope is the rise (up) divided by the run (across) between any two points on the line.

What each symbol means

Slope is a number that tells you how steep a line is. A big slope is a steep hill. A small slope is nearly flat. A negative slope goes downhill as you read left to right.

To find it, pick any two points on the line. See how much the line rises (the change in the up-down value, y) and how much it runs (the change in the left-right value, x). Slope is rise divided by run.

Rise over run: People often say "rise over run". Rise is the change in y (up or down). Run is the change in x (across). Slope is always rise divided by run, never the other way around.

A worked example

Find the slope through (1, 2) and (4, 8)

  1. Label the points: (x_1, y_1) = (1, 2) and (x_2, y_2) = (4, 8).
  2. Write the formula: m = y_2 - y_1x_2 - x_1.
  3. Subtract the y values (rise): 8 - 2 = 6.
  4. Subtract the x values (run): 4 - 1 = 3.
  5. Divide: m = 63 = 2.
  6. The slope is 2: the line rises 2 for every 1 it moves across.

Keep the order matched: Whichever point you call the second one, use it on the top and bottom in the same order. Subtract the y values and the x values in the same direction, or the sign will be wrong.

Why rise over run measures steepness

  1. Steepness means: for each step you take across, how far do you climb?
  2. Between two points, the climb is the change in height, y_2 - y_1 (the rise).
  3. The distance across is the change in the sideways position, x_2 - x_1 (the run).
  4. Climb per step across is climb divided by steps: y_2 - y_1x_2 - x_1. On a straight line this ratio is the same no matter which two points you pick, which is exactly why one number describes the whole line.

Practice questions

Question 1

Find the slope through (0, 0) and (2, 6).

  1. Label the points: (x_1, y_1) = (0, 0) and (x_2, y_2) = (2, 6).
  2. Find the rise (change in y): 6 - 0 = 6.
  3. Find the run (change in x): 2 - 0 = 2.
  4. Divide rise by run: m = 62 = 3.

Answer: 3

Question 2

Find the slope through (1, 5) and (3, 9).

  1. Label the points: (x_1, y_1) = (1, 5) and (x_2, y_2) = (3, 9).
  2. Find the rise (change in y): 9 - 5 = 4.
  3. Find the run (change in x): 3 - 1 = 2.
  4. Divide rise by run: m = 42 = 2.

Answer: 2

Question 3

Find the slope through (2, 7) and (5, 1).

  1. Label the points: (x_1, y_1) = (2, 7) and (x_2, y_2) = (5, 1).
  2. Find the rise (change in y): 1 - 7 = -6.
  3. Find the run (change in x): 5 - 2 = 3.
  4. Divide rise by run: m = -63 = -2.
  5. The slope is negative, so this line goes downhill from left to right.

Answer: -2

Question 4

What is the slope of a flat, horizontal line?

  1. On a horizontal line, the height never changes as you move across.
  2. So the rise (change in y) is 0.
  3. Slope is rise over run, which is 0run = 0.
  4. Every horizontal line has a slope of 0.

Answer: 0

Question 5

A line goes through (4, 4) and (6, 10). Find its slope.

  1. Label the points: (x_1, y_1) = (4, 4) and (x_2, y_2) = (6, 10).
  2. Find the rise (change in y): 10 - 4 = 6.
  3. Find the run (change in x): 6 - 4 = 2.
  4. Divide rise by run: m = 62 = 3.

Answer: 3