The sum of two numbers is $30$.   If we double the larger number, and subtract three times the smaller number, the result is 5.  What is the positive difference between the two numbers?
Let $x$ be the smaller integer and $y$ be the larger integer. We have $x+y=30$ and $2y-3x=5$. Solving for $x$ in terms of $y$ using the first equation, we get $x=30-y$. Now we substitute $x$ in terms of $y$ into the second equation. \begin{align*}
2y-3(30-y)&=5\quad\Rightarrow\\
2y-90+3y&=5\quad\Rightarrow\\
5y&=95\quad\Rightarrow\\
y&=19
\end{align*} And $x=30-19=11$. The positive difference is $y-x=\boxed{8}$.
8
Level 2
Algebra
Problem: The sum of two numbers is $30$.   If we double the larger number, and subtract three times the smaller number, the result is 5.  What is the positive difference between the two numbers?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>Let the two numbers be represented as \( x \) (the larger number) and \( y \) (the smaller number). From the problem, we have the following equations based on the given conditions:

1. The sum of the two numbers:
   \[
   x + y = 30
   \]

2. The condition involving doubling the larger number and subtracting three times the smaller number:
   \[
   2x - 3y = 5
   \]

To solve for \( x \) and \( y \), we can use substitution or elimination. First, we can express \( x \) in terms of \( y \) from the first equation:
\[
x = 30 - y
\]

Next, we substitute this expression for \( x \) into the second equation:
\[
2(30 - y) - 3y = 5
\]

Expanding this gives:
\[
60 - 2y - 3y = 5
\]
\[
60 - 5y = 5
\]

Now, we isolate \( y \):
\[
-5y = 5 - 60
\]
\[
-5y = -55
\]
\[
y = 11
\]

Now that we have \( y \), we can find \( x \) using the first equation:
\[
x = 30 - 11 = 19
\]

Now we have both numbers: \( x = 19 \) and \( y = 11 \). The problem asks for the positive difference between the two numbers:
\[
|x - y| = |19 - 11| = 8
\]

Thus, the final answer is \( 8 \).<|END_OF_LATENT|> The final answer is $8$.