To find the number of upper-level seats ([tex]\( x \)[/tex]) and lower-level seats ([tex]\( y \)[/tex]) in the venue, we need to solve the system of linear equations generated from the ticket pricing and revenue information provided:
1. The revenue from the basketball game:
[tex]\[
20x + 30y = 7000
\][/tex]
2. The revenue from the hockey game:
[tex]\[
15x + 35y = 6500
\][/tex]
We will solve this system of equations step-by-step:
First, to make it easier to eliminate one variable, we can manipulate the equations to make the coefficients of either [tex]\( x \)[/tex] or [tex]\( y \)[/tex] the same.
1. Multiply the first equation by 3 and the second equation by 2 to align the coefficient of [tex]\( x \)[/tex]:
[tex]\[
3(20x + 30y = 7000) \implies 60x + 90y = 21000
\][/tex]
[tex]\[
2(15x + 35y = 6500) \implies 30x + 70y = 13000
\][/tex]
Next, subtract the second modified equation from the first modified equation to eliminate [tex]\( x \)[/tex]:
[tex]\[
(60x + 90y) - (30x + 70y) = 21000 - 13000
\][/tex]
[tex]\[
30x + 20y = 8000
\][/tex]
Now, solve for [tex]\( y \)[/tex]:
Dividing through by 20 to simplify,
[tex]\[
30x + 20y = 8000 \implies y = \frac{8000}{20} \implies y = 400
\][/tex]
So, we have:
[tex]\[
y = 100
\][/tex]
Substitute [tex]\( y = 100 \)[/tex] back into the first original equation to solve for [tex]\( x \)[/tex]:
[tex]\[
20x + 30(100) = 7000
\][/tex]
[tex]\[
20x + 3000 = 7000
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
20x = 7000 - 3000
\][/tex]
[tex]\[
20x = 4000
\][/tex]
[tex]\[
x = \frac{4000}{20} \implies x = 200
\][/tex]
Thus, the venue has [tex]\( 200 \)[/tex] upper-level seats and [tex]\( 100 \)[/tex] lower-level seats.
Therefore, the venue has:
[tex]\[
\boxed{200}
\][/tex]
upper-level seats and
[tex]\[
\boxed{100}
\][/tex]
lower-level seats.
