Consider the lines \begin{align*}
y&=3x+5 \\ 2y&=4x+5 \\ 3y&=9x-2 \\ 2y&=x-3 \\ 4y&=x-5.
\end{align*}Let's say that a pair of lines is $\emph{good}$ if the two lines are either parallel or perpendicular to each other.  Among all the pairs of lines shown, how many pairs are good?
We find the slope for each line. The slopes are $\frac31=3$, $\frac42=2$, $\frac93=3$, $\frac12$, and $\frac14$.
Parallel lines have the same slope, therefore lines $a$ and $c$ are parallel. Perpendicular lines have slopes that are negative reciprocals. None of the slopes above are negative reciprocals, therefore there are no perpendicular lines. There are $1+0=\boxed{1}$ pairs of lines that are parallel or perpendicular.
1
Level 3
Algebra
Problem: Consider the lines \begin{align*}
y&=3x+5 \\ 2y&=4x+5 \\ 3y&=9x-2 \\ 2y&=x-3 \\ 4y&=x-5.
\end{align*}Let's say that a pair of lines is $\emph{good}$ if the two lines are either parallel or perpendicular to each other.  Among all the pairs of lines shown, how many pairs are good?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the number of good pairs of lines, we first need to analyze the slopes of each line. The condition for two lines to be parallel is that they have the same slope, while the condition for them to be perpendicular is that the product of their slopes equals -1.

1. For the first line, y = 3x + 5, the slope is 3.
2. For the second line, 2y = 4x + 5 can be rewritten as y = 2x + 2. The slope is 2.
3. For the third line, 3y = 9x - 2 can be rewritten as y = 3x - 2. The slope is 3.
4. For the fourth line, 2y = x - 3 can be rewritten as y = 0.5x - 1.5. The slope is 0.5.
5. For the fifth line, 4y = x - 5 can be rewritten as y = 0.25x - 1.25. The slope is 0.25.

Next, we list the slopes:
- Line 1: slope = 3
- Line 2: slope = 2
- Line 3: slope = 3
- Line 4: slope = 0.5
- Line 5: slope = 0.25

Now, we check for parallel lines:
- Line 1 and Line 3 are parallel (both have slope 3).
- No other lines share the same slope.

Next, we check for perpendicular lines:
- The product of the slopes of Line 1 (3) and Line 4 (0.5) is 1.5, not -1.
- The product of the slopes of Line 1 (3) and Line 5 (0.25) is 0.75, not -1.
- The product of the slopes of Line 2 (2) and Line 4 (0.5) is 1, not -1.
- The product of the slopes of Line 2 (2) and Line 5 (0.25) is 0.5, not -1.
- The product of the slopes of Line 3 (3) and Line 4 (0.5) is 1.5, not -1.
- The product of the slopes of Line 3 (3) and Line 5 (0.25) is 0.75, not -1.
- The product of the slopes of Line 4 (0.5) and Line 5 (0.25) is 0.125, not -1.

After checking all pairs, the only good pair identified is Line 1 and Line 3, which are parallel. Therefore, the total number of good pairs is 1.<|END_OF_LATENT|> The final answer is $1$.