Let $f$ be a linear function. Compute the slope of $f$ if
$$\int_3^5f(x)dx=0\text{ and }\int_5^7f(x)dx=12.$$
1. Since \( f \) is a linear function, we can write it in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
2. The integral of \( f(x) \) from \( a \) to \( b \) is given by:
   \[
   \int_a^b f(x) \, dx = \int_a^b (mx + b) \, dx = \left[ \frac{m}{2} x^2 + bx \right]_a^b
   \]
3. Given \( \int_3^5 f(x) \, dx = 0 \), we can write:
   \[
   \left[ \frac{m}{2} x^2 + bx \right]_3^5 = 0
   \]
   Evaluating this, we get:
   \[
   \left( \frac{m}{2} (5^2) + b(5) \right) - \left( \frac{m}{2} (3^2) + b(3) \right) = 0
   \]
   Simplifying, we have:
   \[
   \left( \frac{25m}{2} + 5b \right) - \left( \frac{9m}{2} + 3b \right) = 0
   \]
   \[
   \frac{25m}{2} + 5b - \frac{9m}{2} - 3b = 0
   \]
   \[
   \frac{16m}{2} + 2b = 0
   \]
   \[
   8m + 2b = 0 \quad \text{(Equation 1)}
   \]
4. Given \( \int_5^7 f(x) \, dx = 12 \), we can write:
   \[
   \left[ \frac{m}{2} x^2 + bx \right]_5^7 = 12
   \]
   Evaluating this, we get:
   \[
   \left( \frac{m}{2} (7^2) + b(7) \right) - \left( \frac{m}{2} (5^2) + b(5) \right) = 12
   \]
   Simplifying, we have:
   \[
   \left( \frac{49m}{2} + 7b \right) - \left( \frac{25m}{2} + 5b \right) = 12
   \]
   \[
   \frac{49m}{2} + 7b - \frac{25m}{2} - 5b = 12
   \]
   \[
   \frac{24m}{2} + 2b = 12
   \]
   \[
   12m + 2b = 12 \quad \text{(Equation 2)}
   \]
5. We now have the system of equations:
   \[
   \begin{cases}
   8m + 2b = 0 \\
   12m + 2b = 12
   \end{cases}
   \]
6. Subtract Equation 1 from Equation 2:
   \[
   (12m + 2b) - (8m + 2b) = 12 - 0
   \]
   \[
   4m = 12
   \]
   \[
   m = 3
   \]
7. Substitute \( m = 3 \) back into Equation 1 to find \( b \):
   \[
   8(3) + 2b = 0
   \]
   \[
   24 + 2b = 0
   \]
   \[
   2b = -24
   \]
   \[
   b = -12
   \]

The final answer is \( \boxed{ m = 3 } \)