To determine which combinations of new-release movies (at a cost of [tex]$20 each) and classic movies (at a cost of $[/tex]8 each) fit within a budget of [tex]$500, we'll examine each pair of values for \(x\) (new releases) and \(y\) (classics) provided in the problem.
Each pair will be checked to ensure the total cost does not exceed the $[/tex]500 budget.
1. For [tex]\(x = 8\)[/tex] and [tex]\(y = 45\)[/tex]:
- Total cost: [tex]\( 20x + 8y \)[/tex]
- Calculation: [tex]\( 20(8) + 8(45) = 160 + 360 = 520 \)[/tex]
- Since [tex]\(520\)[/tex] exceeds [tex]\(500\)[/tex], this combination is not valid.
2. For [tex]\(x = 10\)[/tex] and [tex]\(y = 22\)[/tex]:
- Total cost: [tex]\( 20x + 8y \)[/tex]
- Calculation: [tex]\( 20(10) + 8(22) = 200 + 176 = 376 \)[/tex]
- Since [tex]\(376\)[/tex] is within [tex]\(500\)[/tex], this combination is valid.
3. For [tex]\(x = 16\)[/tex] and [tex]\(y = 22\)[/tex]:
- Total cost: [tex]\( 20x + 8y \)[/tex]
- Calculation: [tex]\( 20(16) + 8(22) = 320 + 176 = 496 \)[/tex]
- Since [tex]\(496\)[/tex] is within [tex]\(500\)[/tex], this combination is valid.
4. For [tex]\(x = 18\)[/tex] and [tex]\(y = 18\)[/tex]:
- Total cost: [tex]\( 20x + 8y \)[/tex]
- Calculation: [tex]\( 20(18) + 8(18) = 360 + 144 = 504 \)[/tex]
- Since [tex]\(504\)[/tex] exceeds [tex]\(500\)[/tex], this combination is not valid.
After evaluating each combination, the librarian can afford the following:
1. [tex]\( x = 10 \)[/tex] and [tex]\( y = 22 \)[/tex] with a total cost of [tex]\( \$376 \)[/tex]
2. [tex]\( x = 16 \)[/tex] and [tex]\( y = 22 \)[/tex] with a total cost of [tex]\( \$496 \)[/tex]
Therefore, the combinations of [tex]\( x \)[/tex] (new releases) and [tex]\( y \)[/tex] (classics) that the librarian could purchase within the budget are:
[tex]\[
(x = 10, y = 22) \quad \text{and} \quad (x = 16, y = 22)
\][/tex]
