Question

first customer second customer number of chairs 3 5 number of tables 8 2 total cost (in dollars) 74 27 let x be the cost (in dollars) to rent a chair. let y be the cost (in dollars) to rent a table. (a) write a system of equations that could be used to find the rental cost (in dollars) of each chair and each table. \\(\square x + \square y = \square\\) \\(\square x + \square y = \square\\) (b) how much does each chair and each table cost (in dollars) to rent? cost to rent each chair: $\square$ cost to rent each table: $\square$

Explanation:

Part (a)

Step1: Analyze First Customer

The first customer rents 3 chairs (cost \(3x\)) and 8 tables (cost \(8y\)), total cost 74. So equation: \(3x + 8y = 74\).

Step2: Analyze Second Customer

The second customer rents 5 chairs (cost \(5x\)) and 2 tables (cost \(2y\)), total cost 27. So equation: \(5x + 2y = 27\).

Step1: Solve the system. Let's use elimination. Multiply the second equation by 4: \(20x + 8y = 108\).

Step2: Subtract the first equation (\(3x + 8y = 74\)) from the new second equation: \((20x + 8y) - (3x + 8y) = 108 - 74\) → \(17x = 34\) → \(x = 2\).

Step3: Substitute \(x = 2\) into \(5x + 2y = 27\): \(5(2) + 2y = 27\) → \(10 + 2y = 27\) → \(2y = 17\) → \(y = 8.5\)? Wait, no, wait. Wait, 52=10, 27-10=17? Wait, no, 5x + 2y =27, x=2: 10 + 2y=27 → 2y=17? Wait, that can't be. Wait, no, I made a mistake. Wait, first equation: 3x +8y=74, x=2: 6 +8y=74 → 8y=68 → y=8.5? But second customer: 52 +2*8.5=10 +17=27. Oh right! So that's correct. Wait, but let's check again. First customer: 3 chairs at $2: $6, 8 tables at $8.5: $68, total $74. Correct. Second customer: 5 chairs at $2: $10, 2 tables at $8.5: $17, total $27. Correct.

Wait, but maybe I messed up the elimination. Let's do it again. The two equations:

1. \(3x + 8y = 74\)
2. \(5x + 2y = 27\)

Let's solve equation 2 for \(y\): \(2y = 27 -5x\) → \(y = \frac{27 -5x}{2}\). Substitute into equation 1:

\(3x + 8(\frac{27 -5x}{2}) = 74\) → \(3x + 4(27 -5x) =74\) → \(3x +108 -20x =74\) → \(-17x = -34\) → \(x=2\). Then \(y = \frac{27 -10}{2} = \frac{17}{2}=8.5\). Wait, but the problem is about renting chairs and tables, maybe decimal is okay. Wait, but let's check the arithmetic again. 5x +2y=27, x=2: 10 +2y=27 → 2y=17 → y=8.5. Yes. So chair is $2, table is $8.5? Wait, but maybe I made a mistake in the problem setup. Wait, first customer: 3 chairs, 8 tables, total 74. 32=6, 88.5=68, 6+68=74. Correct. Second customer: 52=10, 28.5=17, 10+17=27. Correct. So that's the solution.

Answer:

\(3x + 8y = 74\)
\(5x + 2y = 27\)

Part (b)