problem: What are the coordinates of the point on the line $5x-9y=42$ such that the $x$ and $y$ coordinates are the additive inverses of each other? Express your answer as an ordered pair.
solution: Our point lies on $5x-9y=42$ with the condition that $x=-y$. Thus, we have the system of equations \begin{align*}
  5x-9y &= 42\\
  x &= -y.
  \end{align*} Substituting $x= -y$ into the first equation gives  \begin{align*}
  5(-y) -9y &=42\\
  \Rightarrow -14y &= 42\\
  \Rightarrow y &=-3.
  \end{align*} Thus $x = -y = -(-3) = 3$, so our desired point is $\boxed{(3,-3)}$.