To determine which values of [tex]\( x \)[/tex] (number of new-release movies) and [tex]\( y \)[/tex] (number of classic movies) the librarian could have bought within a budget of \[tex]$500, we need to calculate the total cost for each combination provided and check if it fits into the budget.
The cost formulas are:
- New-release movie: \(\$[/tex]20 \times x\)
- Classic movie: [tex]\(\$8 \times y\)[/tex]
1. For the combination [tex]\( (x = 8, y = 45) \)[/tex]:
[tex]\[
\text{Total Cost} = 20 \times 8 + 8 \times 45
\][/tex]
[tex]\[
\text{Total Cost} = 160 + 360 = 520
\][/tex]
Since \[tex]$520 exceeds the \$[/tex]500 budget, this combination is not valid.
2. For the combination [tex]\( (x = 10, y = 22) \)[/tex]:
[tex]\[
\text{Total Cost} = 20 \times 10 + 8 \times 22
\][/tex]
[tex]\[
\text{Total Cost} = 200 + 176 = 376
\][/tex]
Since \[tex]$376 is within the \$[/tex]500 budget, this combination is valid.
3. For the combination [tex]\( (x = 16, y = 22) \)[/tex]:
[tex]\[
\text{Total Cost} = 20 \times 16 + 8 \times 22
\][/tex]
[tex]\[
\text{Total Cost} = 320 + 176 = 496
\][/tex]
Since \[tex]$496 is within the \$[/tex]500 budget, this combination is also valid.
4. For the combination [tex]\( (x = 18, y = 18) \)[/tex]:
[tex]\[
\text{Total Cost} = 20 \times 18 + 8 \times 18
\][/tex]
[tex]\[
\text{Total Cost} = 360 + 144 = 504
\][/tex]
Since \[tex]$504 exceeds the \$[/tex]500 budget, this combination is not valid.
Hence, the valid combinations of the number of new-release and classic movies bought within the \$500 budget are:
[tex]\[
(x = 10, y = 22) \quad \text{and} \quad (x = 16, y = 22)
\][/tex]
