The problem states that $y$ varies directly as $x$, meaning that the relationship between $y$ and $x$ can be written as:

$y = kx$

where $k$ is the constant of variation (also called the constant of proportionality).

Let's solve each problem step by step.

1. If $y = 12$ when $x = 4$, find $y$ when $x = 12$.

First, find the constant $k$: $12 = k \times 4 \quad \Rightarrow \quad k = \frac{12}{4} = 3$ Now, use $k = 3$ to find $y$ when $x = 12$: $y = 3 \times 12 = 36$

2. If $y = 25$ when $x = 5$, find $x$ when $y = 100$.

Find the constant $k$: $25 = k \times 5 \quad \Rightarrow \quad k = \frac{25}{5} = 5$ Now, use $k = 5$ to find $x$ when $y = 100$: $100 = 5 \times x \quad \Rightarrow \quad x = \frac{100}{5} = 20$

3. If $y = -8$ when $x = 2$, find $x$ when $y = 24$.

Find the constant $k$: $-8 = k \times 2 \quad \Rightarrow \quad k = \frac{-8}{2} = -4$ Now, use $k = -4$ to find $x$ when $y = 24$: $24 = -4 \times x \quad \Rightarrow \quad x = \frac{24}{-4} = -6$

4. If $y = 14$ when $x = 7$, find $y$ when $x = 21$.

Find the constant $k$: $14 = k \times 7 \quad \Rightarrow \quad k = \frac{14}{7} = 2$ Now, use $k = 2$ to find $y$ when $x = 21$: $y = 2 \times 21 = 42$

5. If $y = -18$ when $x = -6$, find $x$ when $y = 90$.

Find the constant $k$: $-18 = k \times -6 \quad \Rightarrow \quad k = \frac{-18}{-6} = 3$ Now, use $k = 3$ to find $x$ when $y = 90$: $90 = 3 \times x \quad \Rightarrow \quad x = \frac{90}{3} = 30$

Summary of Answers:

1. $y = 36$
2. $x = 20$
3. $x = -6$
4. $y = 42$
5. $x = 30$

Would you like more details or have any questions?

Here are five related questions for further practice:

1. What happens to $y$ if $x$ is halved, assuming direct variation?
2. If $y$ varies directly as $x$ and $y = 18$ when $x = 6$, what is $y$ when $x = 15$?
3. How do you determine the constant of proportionality in a direct variation problem?
4. If the constant $k$ is negative, what effect does this have on the relationship between $y$ and $x$?
5. If $y = 50$ when $x = 10$, what is the formula for the relationship between $y$ and $x$?

Tip: In problems of direct variation, once you find the constant $k$, solving for any other values becomes a straightforward application of the formula $y = kx$.