### Instruction: Find the value of $k$ for which $kx^2 -5x-12 = 0$ has solutions $x=3$ and $ x = -\frac{4}{3}$.### Response: To find the value of $k$, I can use the fact that if $x=3$ and $ x = -\frac{4}{3}$ are solutions of the quadratic equation, then they must satisfy it when I plug them in.
So I can write two equations: $k(3)^2 - 5(3) - 12 = 0$ and $k(-\frac{4}{3})^2 - 5(-\frac{4}{3}) - 12 = 0$.
Simplifying the first equation, I get $9k - 27 = 0$, so $k = 3$.
Simplifying the second equation, I get $\frac{16}{9}k + \frac{20}{3} - 12 = 0$, so $\frac{16}{9}k = \frac{16}{3}$, so $k = 3$.
Both equations give me the same value of $k$, which is good.
Therefore, the value of $k$ for which $kx^2 -5x-12 = 0$ has solutions $x=3$ and $ x = -\frac{4}{3}$ is $k=3$.