Use Matrices to Represent Systems of Linear Equations

Let $a$, $b$, $c$, $x$, $y$, and $z$ be real numbers that satisfy the three equations $$\begin{align*} 13x + by + cz &= 0 \\ ax + 23y + cz &= 0 \\ ax + by + 42z &= 0. \end{align*}$$Suppose that $a \ne 13$ and $x \ne 0$. What is the value of $$\frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c - 42} ?$$

(Hint: The determinant of the matrix $\begin{bmatrix}13 & b& c \\ a& 23& c \\ a& b &42\end{bmatrix}$ is zero. Now try playing with row operations to get $13/(a-3)$, etc. to pop up.)

Source: Math Prize for Girls