Solved Direct and Inverse Variation Worksheet Name: Find the ... - Free Printable

Let's solve each problem step by step using the concepts of direct and inverse variation.

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🔹 Direct Variation:

If $ y $ varies directly with $ x $, then:
$$
y = kx
$$
where $ k $ is the constant of variation.

🔹 Inverse Variation:

If $ y $ varies inversely with $ x $, then:
$$
y = \frac{k}{x} \quad \text{or} \quad xy = k
$$

We'll use these formulas to solve each problem.

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1) $ y $ varies directly with $ x $. If $ y = -4 $ when $ x = 2 $, find $ y $ when $ x = -6 $.

Step 1: Use direct variation: $ y = kx $

Given: $ y = -4 $, $ x = 2 $

$$
-4 = k(2) \Rightarrow k = \frac{-4}{2} = -2
$$

So the equation is: $ y = -2x $

Step 2: Find $ y $ when $ x = -6 $

$$
y = -2(-6) = 12
$$

✔ Answer: $ y = 12 $

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2) $ y $ varies inversely with $ x $. If $ y = 40 $ when $ x = 16 $, find $ x $ when $ y = -5 $.

Step 1: Use inverse variation: $ y = \frac{k}{x} $ or $ xy = k $

Given: $ y = 40 $, $ x = 16 $

$$
k = xy = 40 \times 16 = 640
$$

So $ xy = 640 $

Step 2: Find $ x $ when $ y = -5 $

$$
x(-5) = 640 \Rightarrow x = \frac{640}{-5} = -128
$$

✔ Answer: $ x = -128 $

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3) $ y $ varies inversely with $ x $. If $ y = 7 $ when $ x = -4 $, find $ y $ when $ x = 5 $.

Step 1: Use $ xy = k $

Given: $ y = 7 $, $ x = -4 $

$$
k = (-4)(7) = -28
$$

So $ xy = -28 $

Step 2: Find $ y $ when $ x = 5 $

$$
5y = -28 \Rightarrow y = \frac{-28}{5} = -5.6
$$

✔ Answer: $ y = -5.6 $ (or $ -\frac{28}{5} $)

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4) $ y $ varies directly with $ x $. If $ y = 15 $ when $ x = -18 $, find $ y $ when $ x = 1.6 $.

Step 1: Use $ y = kx $

Given: $ y = 15 $, $ x = -18 $

$$
15 = k(-18) \Rightarrow k = \frac{15}{-18} = -\frac{5}{6}
$$

So $ y = -\frac{5}{6}x $

Step 2: Find $ y $ when $ x = 1.6 $

$$
y = -\frac{5}{6}(1.6) = -\frac{5}{6} \times \frac{8}{5} = -\frac{40}{30} = -\frac{4}{3} \approx -1.333...
$$

Wait — let's compute it carefully:

$ 1.6 = \frac{8}{5} $, so:

$$
y = -\frac{5}{6} \times \frac{8}{5} = -\frac{40}{30} = -\frac{4}{3} \approx -1.333
$$

✔ Answer: $ y = -\frac{4}{3} $ or $ -1.\overline{3} $

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5) $ y $ varies directly with $ x $. If $ y = 75 $ when $ x = 25 $, find $ x $ when $ y = 25 $.

Step 1: Use $ y = kx $

Given: $ y = 75 $, $ x = 25 $

$$
75 = k(25) \Rightarrow k = \frac{75}{25} = 3
$$

So $ y = 3x $

Step 2: Find $ x $ when $ y = 25 $

$$
25 = 3x \Rightarrow x = \frac{25}{3} \approx 8.333...
$$

✔ Answer: $ x = \frac{25}{3} $ or $ 8.\overline{3} $

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✔ Final Answers:

1) $ y = 12 $
2) $ x = -128 $
3) $ y = -5.6 $ or $ -\frac{28}{5} $
4) $ y = -\frac{4}{3} $ or $ -1.\overline{3} $
5) $ x = \frac{25}{3} $ or $ 8.\overline{3} $

Let me know if you'd like these as decimals or fractions!