The numbers $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 42, $x$ is twice $y$. What is the value of $y$ when $x=-8$?
We know that when $x+y=42$, $x=2y$. Substituting $2y$ in for $x$ in the first equation gives $3y=42$, or $y=14$. The value of $x$ is then $2(14)=28$. Since $x$ and $y$ are inversely proportional, the product $xy$ is constant. Let $xy=k$. When $x=28$ and $y=14$, $k=(28)(14)=392$. Therefore, when $x=-8$, $(-8)y=392$, giving $y=\boxed{-49}$.
-49
Level 4
Algebra
Problem: The numbers $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 42, $x$ is twice $y$. What is the value of $y$ when $x=-8$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the problem, we start by understanding the relationship between the variables $x$ and $y$. Since $x$ and $y$ are inversely proportional, we can express this relationship as $x \cdot y = k$, where $k$ is a constant.

Next, we analyze the conditions given in the problem. We know that the sum of $x$ and $y$ is 42, which gives us the equation:

1. $x + y = 42$.

Additionally, we are told that $x$ is twice $y$, leading to the equation:

2. $x = 2y$.

We can substitute equation 2 into equation 1 to find the values of $x$ and $y$. Replacing $x$ in the first equation gives:

3. $2y + y = 42$.

This simplifies to:

4. $3y = 42$.

Dividing both sides by 3 yields:

5. $y = 14$.

Now, substituting $y = 14$ back into equation 2 to find $x$:

6. $x = 2(14) = 28$.

At this point, we have determined that when the sum of $x$ and $y$ is 42, $x$ equals 28 and $y$ equals 14. 

Next, we need to find the value of $y$ when $x = -8$. Since $x$ and $y$ are inversely proportional, we can use the constant $k$ we derived earlier. We calculate $k$ using the values we found:

7. $k = x \cdot y = 28 \cdot 14 = 392$.

Now, we can find $y$ when $x = -8$ by using the relationship $x \cdot y = k$:

8. $-8 \cdot y = 392$.

To isolate $y$, we divide both sides by -8:

9. $y = \frac{392}{-8}$.

Calculating this gives:

10. $y = -49$.

Thus, the value of $y$ when $x = -8$ is $-49$.<|END_OF_LATENT|> The final answer is $-49$.