Algebra I

Direct and Inverse Variation

Lesson

Two quantities often grow or shrink together. Variation captures the simplest two patterns.

Direct variation

y = kx

As $x$ doubles, so does $y$ . The graph is a straight line through the origin with slope $k$ . The constant $k$ is the constant of variation.

Inverse variation

y = \frac{k}{x} \quad\text{or}\quad xy = k

As $x$ doubles, $y$ is cut in half. Their product is constant.

Worked example 1 — direct

$y$ varies directly with $x$ , and $y = 20$ when $x = 5$ .

k = y / x = 20 / 5 = 4

The equation is $y = 4x$ .

Worked example 2 — inverse

$y$ varies inversely with $x$ , and $y = 6$ when $x = 4$ .

k = xy = 4 \cdot 6 = 24

The equation is $y = 24 / x$ . To find $y$ when $x = 8$ : $y = 24 / 8 = 3$ .

Spot the type

If $y / x$ is constant, it’s direct.
If $x \cdot y$ is constant, it’s inverse.

Real-world cue: direct shows up in pay-per-hour, mass-per-volume, cost-per-item. Inverse shows up in speed-vs-time at fixed distance, force-vs-distance, and intensity-vs-distance.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

y \text{ varies directly with } x; \ y = 12 \text{ when } x = 3. \ k = ?

Problem 2

y \text{ varies directly with } x; \ y = 20 \text{ when } x = 5. \ k = ?

Problem 3

y \text{ varies inversely with } x; \ y = 6 \text{ when } x = 4. \ k = ?

Problem 4

y \text{ varies inversely with } x; \ y = 5 \text{ when } x = 2. \ k = ?

Practice

Standard problems matching the lesson.

Problem 5

y = kx \text{ with } y = 15 \text{ at } x = 3. \ k = ?

Problem 6

Direct: y=24 at x=8. y at x=10?

Problem 7

Direct: y=10 at x=2. y at x=7?

Problem 8

Direct: y=18 at x=6. x at y=27?

Problem 9

\text{Inverse: } xy = k. \ y = 12, \ x = 3. \ k = ?

Problem 10

Inverse: y=8 at x=5. y at x=10?

Problem 11

Inverse: y=20 at x=2. y at x=8?

Problem 12

Inverse: y=6 at x=4. x at y=8?

Problem 13

Direct: y=4 at x=1. y at x=10?

Problem 14

Inverse: y=2 at x=15. y at x=6?

Problem 15

Pay = k * hours: 60 dollars for 4 hours. Pay for 7?

Problem 16

Time = k/speed. 60 mph * 3 hrs. Time at 90 mph?

Problem 17

\text{Direct: } y = 21 \text{ at } x = 7. \ k = ?

Problem 18

\text{Inverse: } y = 9 \text{ at } x = 4. \ k = ?

Challenge

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

Problem 19

Inverse: y=2 at x=18. y at x=12?

Problem 20

\text{Direct: } y = -12 \text{ at } x = 4. \ k = ?

Problem 21

Direct: k=2/3. y at x=9?

Problem 22

\text{Inverse: } y = 10 \text{ at } x = 0.5. \ k = ?

Problem 23

F * d = k. F=100 at d=2. F at d=5?

Problem 24

Direct variation y = kx. y at x = 0?

Problem 25

Car: 240 mi at 60 mph = 4 hr. Hours at 80 mph?

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