Answer
$\text{Orchestra Seats=100}$;
$\text {Main seats =210}$,
and
$\text{Balcony Seats =190}$
Work Step by Step
Let us consider that
$\text{Orchestra Seats=x}$ ;$\text {Main seats =y}$, and $\text{Balcony Seats = z}$
We set the system of equations as follows:
$x+y+z=500~~~~(1) \\ 50x+35y+25z=17100~~~~(2) \\ \dfrac{50}{2} x+35y+25z=14600~~~(3)$
Multiply equation (1) by $-50$ and then add the new equation to equation (2) to eliminate $x$.
Thus, we get:
$-15y-25z=-7900~~~~~(4)$
Next, multiply equation (1) by $-25$ and then add the new equation to equation (3) to solve for $y$.
Thus, we get:
$10y=2100 \implies y=210$
Now, back-substitute the value of $y$ into equation (4) to solve for $z$.
$-15(210)-25z=-7900 \implies z=190$
Finally, back-substitute the value of $y,z$ into equation (1) to solve for $x$.
$x+210+190=500 \implies x=100$
So, we have:
$\text{Orchestra Seats=100}$ ;$\text {Main seats =210}$, and $\text{Balcony Seats =190}$
