Answer
$8$ TV ads
$30$ radio ads
$22$ newspaper ads
Work Step by Step
We have to solve the system:
$$\begin{align*}
\begin{cases}
x+y+z&=60\quad&\text{Equation }1\\
1000x+200y+500z&=25,000\quad&\text{Equation }2\\
y&=x+z\quad&\text{Equation }3.
\end{cases}
\end{align*}$$
We rewrite the system as a linear system in $\textit{two}$ variables by substituting $x+z$ for $y$ in Equations $1$ and $2$:
$$\begin{align*}
x+y+z&=60\quad\text{Write Equation }1.\\
x+(x+z)+z&=60\quad\text{Substitute }x+z \text{ to }y.\\
2x+2z&=60\quad\text{New Equation }1.\\\\
1000x+200y+500z&=25,000\quad\text{Write Equation }2.\\
1000x+200(x+z)+500z&=25,000\quad\text{Substitute }x+z \text{ to }y.\\
1200x+700z&=25,000\quad\text{New Equation }2.
\end{align*}$$
We solve the new linear system for both its variables:
$$\begin{align*}
-1200x-1200z&=-36,000\quad\text{Add }-600\text{ times new Equation }1\\
120x+700z&=25,000\quad\text{to new Equation }2.\\
\text{___________}&\text{______}\\
-500z&=-11,000\\
z&=22\quad\text{Solve for }z.\\
x&=8\quad\text{Substitute into new Equation }1\text{ to find }x.\\
y&=30\quad\text{Substitute into Equation }3\text{ to find }y.
\end{align*}$$
The solution is $x=8, y=30,z=22$ or the ordered triple $(8,30,22)$. So the department should run $8$ TV ads, $30$ radio ads and $22$ newspaper ads each month.
Note: It is recommended to check the solution in each of the original equations.
