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Algebra I

Slope from Two Points

Lesson

The slope of a line measures how steep it is — how much it rises (or falls) for each unit you move to the right. We usually call slope mm.

Given two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the slope is the rise over the run:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

The numerator (top) is the change in yy — the rise. The denominator (bottom) is the change in xx— the run. As long as you subtract in the same order top and bottom, you can pick either point as “point 1”.

Worked example 1

Find the slope between (1,2)(1, 2) and (4,8)(4, 8).

m=8−24−1=63=2m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2

Worked example 2

Find the slope between (−1,5)(-1, 5) and (2,−1)(2, -1).

m=−1−52−(−1)=−63=−2m = \frac{-1 - 5}{2 - (-1)} = \frac{-6}{3} = -2

The negative slope means the line goes down from left to right.

How to type your answer

Just type the number. For fractions, type as a fraction like 3/2 or as a decimal like 1.5.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

(0,0) and (1,2)(0, 0) \text{ and } (1, 2)

Problem 2

(0,0) and (1,3)(0, 0) \text{ and } (1, 3)

Problem 3

(1,1) and (2,2)(1, 1) \text{ and } (2, 2)

Problem 4

(0,1) and (1,5)(0, 1) \text{ and } (1, 5)

Practice

Standard problems matching the lesson.

Problem 5

(1,2) and (4,8)(1, 2) \text{ and } (4, 8)

Problem 6

(0,1) and (3,7)(0, 1) \text{ and } (3, 7)

Problem 7

(2,5) and (6,13)(2, 5) \text{ and } (6, 13)

Problem 8

(1,4) and (4,1)(1, 4) \text{ and } (4, 1)

Problem 9

(−1,5) and (2,−1)(-1, 5) \text{ and } (2, -1)

Problem 10

(0,0) and (5,15)(0, 0) \text{ and } (5, 15)

Problem 11

(−2,−3) and (1,6)(-2, -3) \text{ and } (1, 6)

Problem 12

(3,8) and (5,2)(3, 8) \text{ and } (5, 2)

Problem 13

(−4,2) and (0,2)(-4, 2) \text{ and } (0, 2)

Problem 14

(2,−1) and (5,8)(2, -1) \text{ and } (5, 8)

Problem 15

(−3,4) and (1,−4)(-3, 4) \text{ and } (1, -4)

Problem 16

(0,−5) and (2,1)(0, -5) \text{ and } (2, 1)

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

(−3,−1) and (1,7)(-3, -1) \text{ and } (1, 7)

Problem 18

(−2,5) and (3,−5)(-2, 5) \text{ and } (3, -5)

Problem 19

(−5,−2) and (−1,6)(-5, -2) \text{ and } (-1, 6)

Problem 20

(4,−3) and (−2,9)(4, -3) \text{ and } (-2, 9)

Problem 21

(−4,5) and (2,−7)(-4, 5) \text{ and } (2, -7)

Problem 22

(1,−10) and (4,5)(1, -10) \text{ and } (4, 5)

Practice

Standard problems matching the lesson.

Problem 23

A taxi fare is 9 dollars after 2 miles and 15 dollars after 5 miles. Find the per-mile rate (dollars per mile).

Problem 24

A pool has 80 gallons at minute 0 and 60 gallons at minute 4. Find the rate of change (gallons per minute, signed).

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

A drone is at 30 m altitude at second 1, and at 18 m at second 5. Find the rate of change (meters per second, signed).

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