Direct and Inverse Variation Worksheet with five problems to solve.

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.

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🔹 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 $.

Since it's direct variation, use:
$$
y = kx
$$

Plug in $ y = -4 $, $ x = 2 $ to find $ k $:
$$
-4 = k(2) \Rightarrow k = \frac{-4}{2} = -2
$$

Now use $ k = -2 $ to 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 $.

Use inverse variation formula:
$$
y = \frac{k}{x} \Rightarrow k = xy
$$

Find $ k $ using $ y = 40 $, $ x = 16 $:
$$
k = (40)(16) = 640
$$

Now use $ k = 640 $ and $ y = -5 $ to find $ x $:
$$
-5 = \frac{640}{x} \Rightarrow x = \frac{640}{-5} = -128
$$

✔ Answer: $ x = -128 $

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

Again, $ y = \frac{k}{x} $, so $ k = xy $

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

Now use $ x = 5 $:
$$
y = \frac{-28}{5} = -5.6
$$

✔ Answer: $ y = -5.6 $ or $ y = -\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 $.

Direct variation: $ y = kx $

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

Now find $ y $ when $ x = 1.6 $:
$$
y = \left(-\frac{5}{6}\right)(1.6)
$$

Convert 1.6 to fraction: $ 1.6 = \frac{8}{5} $

So:
$$
y = -\frac{5}{6} \cdot \frac{8}{5} = -\frac{40}{30} = -\frac{4}{3} \approx -1.333...
$$

✔ Answer: $ y = -\frac{4}{3} $ or $ y \approx -1.33 $

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

Direct variation: $ y = kx $

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

Now use $ y = 25 $:
$$
25 = 3x \Rightarrow x = \frac{25}{3} \approx 8.333...
$$

✔ Answer: $ x = \frac{25}{3} $ or $ x \approx 8.33 $

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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!