Direct Variation | Math Coach Amy

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Key Ideas Easy Medium Hard

y = kx

k = yx

What to Remember

The formula

Direct variation means y = kx.

k is the constant of variation.

Find k first

Rearrange to k = yx.

Substitute the given pair, then reduce.

A square

If y varies directly with the square of x, the formula is y = kx².

Then k = yx².

Use the new x

Write y = kx with the value of k you found.

Substitute the new x and simplify.

Level 1

Easy

y = kx

Example

y = 20 when x = 4. Find y when x = 7.

y = kx

k = yx

k = 204

k = 5

y = 5x

y = 5×7

y = 35

Answer: y = 35

y = 18 when x = 6. Find y when x = 10.

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y = kx

k = yx

k = 186

k = 3

y = 3x

y = 3×10

y = 30

Answer: y = 30

y = 12 when x = 3. Find y when x = 8.

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y = kx

k = yx

k = 123

k = 4

y = 4x

y = 4×8

y = 32

Answer: y = 32

y = 28 when x = 7. Find y when x = 5.

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y = kx

k = yx

k = 287

k = 4

y = 4x

y = 4×5

y = 20

Answer: y = 20

y = 45 when x = 9. Find y when x = 4.

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y = kx

k = yx

k = 459

k = 5

y = 5x

y = 5×4

y = 20

Answer: y = 20

y = 16 when x = 2. Find y when x = 9.

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y = kx

k = yx

k = 162

k = 8

y = 8x

y = 8×9

y = 72

Answer: y = 72

Level 2

Medium

y varies directly with the square of x

Example

y varies directly with the square of x. y = 12 when x = 2. Find y when x = 5.

y = kx²

k = yx²

x² = 4

k = 124

k = 3

y = 3x²

x² = 25

y = 3×25

y = 75

Answer: y = 75

y varies directly with the square of x. y = 18 when x = 3. Find y when x = 2.

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y = kx²

k = yx²

x² = 9

k = 189

k = 2

y = 2x²

x² = 4

y = 2×4

y = 8

Answer: y = 8

y varies directly with the square of x. y = 32 when x = 4. Find y when x = 3.

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y = kx²

k = yx²

x² = 16

k = 3216

k = 2

y = 2x²

x² = 9

y = 2×9

y = 18

Answer: y = 18

y varies directly with the square of x. y = 20 when x = 2. Find y when x = 6.

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y = kx²

k = yx²

x² = 4

k = 204

k = 5

y = 5x²

x² = 36

y = 5×36

y = 180

Answer: y = 180

y varies directly with the square of x. y = 48 when x = 4. Find y when x = 1.

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y = kx²

k = yx²

x² = 16

k = 4816

k = 3

y = 3x²

x² = 1

y = 3×1

y = 3

Answer: y = 3

y varies directly with the square of x. y = 50 when x = 5. Find y when x = 2.

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y = kx²

k = yx²

x² = 25

k = 5025

k = 2

y = 2x²

x² = 4

y = 2×4

y = 8

Answer: y = 8

Level 3

Hard

A direct-variation situation

Example

Distance varies directly with time. A car travels 150 miles in 3 hours. How far does it travel in 5 hours?

d = kt

k = dt

k = 1503

k = 50

d = 50t

d = 50×5

d = 250

The distance is 250 miles.

Answer: The distance is 250 miles.

The cost of notebooks varies directly with the number bought. 4 notebooks cost 12 dollars. What is the cost of 7 notebooks?

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c = kn

k = cn

k = 124

k = 3

c = 3n

c = 3×7

c = 21

The cost is 21 dollars.

Answer: The cost is 21 dollars.

The weight of a wire varies directly with its length. 6 feet of wire weighs 18 ounces. What is the weight of 10 feet?

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w = kn

k = wn

k = 186

k = 3

w = 3n

w = 3×10

w = 30

The weight is 30 ounces.

Answer: The weight is 30 ounces.

Pay varies directly with hours worked. A worker earns 48 dollars in 6 hours. How much does the worker earn in 9 hours?

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p = kh

k = ph

k = 486

k = 8

p = 8h

p = 8×9

p = 72

The pay is 72 dollars.

Answer: The pay is 72 dollars.

Water from a hose varies directly with the minutes it runs. 8 minutes uses 24 gallons. How many gallons are used in 5 minutes?

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g = km

k = gm

k = 248

k = 3

g = 3m

g = 3×5

g = 15

The amount is 15 gallons.

Answer: The amount is 15 gallons.

Calories burned vary directly with minutes of running. 20 minutes burns 160 calories. How many calories are burned in 15 minutes?

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c = km

k = cm

k = 16020

k = 8

c = 8m

c = 8×15

c = 120

The amount is 120 calories.

Answer: The amount is 120 calories.

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