Find the missing variable in direct and inverse variation problems.

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

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

Direct variation means:
$$
y = kx
$$
where $k$ is the constant of variation.

Step 1: Find $k$ using the given values:
$$
-4 = k \cdot 2 \Rightarrow k = \frac{-4}{2} = -2
$$

Step 2: Use $k = -2$ to find $y$ when $x = -6$:
$$
y = -2 \cdot (-6) = 12
$$

✔ Answer: $y = 12$

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

Inverse variation means:
$$
y = \frac{k}{x}
\quad \text{or} \quad xy = k
$$

Step 1: Find $k$ using $y = 40$, $x = 16$:
$$
k = x \cdot y = 16 \cdot 40 = 640
$$

Step 2: Use $k = 640$ to find $x$ when $y = -5$:
$$
xy = 640 \Rightarrow x \cdot (-5) = 640 \Rightarrow x = \frac{640}{-5} = -128
$$

✔ Answer: $x = -128$

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

Again, inverse variation: $xy = k$

Step 1: Find $k$:
$$
k = x \cdot y = (-4) \cdot 7 = -28
$$

Step 2: Use $k = -28$ to find $y$ when $x = 5$:
$$
5 \cdot y = -28 \Rightarrow y = \frac{-28}{5} = -5.6
$$

✔ 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.

Direct variation: $y = kx$

Step 1: Find $k$:
$$
15 = k \cdot (-18) \Rightarrow k = \frac{15}{-18} = -\frac{5}{6}
$$

Step 2: Use $k = -\frac{5}{6}$ to find $y$ when $x = 1.6$:
Note: $1.6 = \frac{8}{5}$

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

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

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

Direct variation: $y = kx$

Step 1: Find $k$:
$$
75 = k \cdot 25 \Rightarrow k = \frac{75}{25} = 3
$$

Step 2: Use $k = 3$ to find $x$ when $y = 25$:
$$
25 = 3x \Rightarrow x = \frac{25}{3} \approx 8.333...
$$

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

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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 in decimal or fraction form only!