A party rental company has chairs and tables for rent. The total cost to rent 5 chairs and 6 tables is \( \$ 53 \). The total cost to rent 3 chairs and 2 tables is \( \$ 21 \). What is the cost to rent each chair and each table? Cost to rent each chair: \( \$ \) Cost to rent each table: \( \$ \)

Let's denote the cost to rent each chair as \( c \) dollars and the cost to rent each table as \( t \) dollars. From the given information, we can set up the following system of equations: 1. For 5 chairs and 6 tables: \( 5c + 6t = 53 \) 2. For 3 chairs and 2 tables: \( 3c + 2t = 21 \) We can solve this system of equations to find the values of \( c \) and \( t \), which represent the cost to rent each chair and each table, respectively. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}5c+6t=53\\3c+2t=21\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}c=\frac{53-6t}{5}\\3c+2t=21\end{array}\right.\) - step2: Substitute the value of \(c:\) \(3\times \frac{53-6t}{5}+2t=21\) - step3: Simplify: \(\frac{3\left(53-6t\right)}{5}+2t=21\) - step4: Multiply both sides of the equation by LCD: \(\left(\frac{3\left(53-6t\right)}{5}+2t\right)\times 5=21\times 5\) - step5: Simplify the equation: \(159-8t=105\) - step6: Move the constant to the right side: \(-8t=105-159\) - step7: Subtract the numbers: \(-8t=-54\) - step8: Change the signs: \(8t=54\) - step9: Divide both sides: \(\frac{8t}{8}=\frac{54}{8}\) - step10: Divide the numbers: \(t=\frac{27}{4}\) - step11: Substitute the value of \(t:\) \(c=\frac{53-6\times \frac{27}{4}}{5}\) - step12: Simplify: \(c=\frac{5}{2}\) - step13: Calculate: \(\left\{ \begin{array}{l}c=\frac{5}{2}\\t=\frac{27}{4}\end{array}\right.\) - step14: Check the solution: \(\left\{ \begin{array}{l}c=\frac{5}{2}\\t=\frac{27}{4}\end{array}\right.\) - step15: Rewrite: \(\left(c,t\right) = \left(\frac{5}{2},\frac{27}{4}\right)\) The solution to the system of equations is \( c = 2.5 \) and \( t = 6.75 \). Therefore, the cost to rent each chair is $2.50 and the cost to rent each table is $6.75.