Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Question

A realtor is decorating some homes for sale, putting a certain number of decorative pillows on each twin bed and a certain number on each queen bed. In one house, she decorated 2 twin beds and 2 queen beds and used a total of 34 pillows. At another house, she used 22 pillows to spruce up 2 twin beds and 1 queen bed. How many decorative pillows did the realtor arrange on each bed?

The realtor used
pillows on every twin bed and
pillows on every queen bed.

Answer 1

Let's let \( x \) represent the number of decorative pillows on each twin bed and \( y \) represent the number of decorative pillows on each queen bed. We can then set up the system of equations based on the information given in the problem.

1. For the first house where the realtor decorated 2 twin beds and 2 queen beds using a total of 34 pillows, we can write the equation: \[ 2x + 2y = 34 \]

2. For the second house where the realtor used 22 pillows to decorate 2 twin beds and 1 queen bed, the equation is: \[ 2x + 1y = 22 \]

Now we have the system of equations: \[ \begin{cases} 2x + 2y = 34 \quad (1)\ 2x + y = 22 \quad (2) \end{cases} \]

We can simplify equation (1) by dividing everything by 2: \[ x + y = 17 \quad (3) \]

Now we have a simpler system: \[ \begin{cases} x + y = 17 \quad (3)\ 2x + y = 22 \quad (2) \end{cases} \]

Next, we can solve these equations. We can subtract equation (3) from equation (2): \[ (2x + y) - (x + y) = 22 - 17 \] This simplifies to: \[ 2x - x = 5 \] \[ x = 5 \]

Now that we have \( x \), we can substitute it back into equation (3) to find \( y \): \[ 5 + y = 17 \] \[ y = 17 - 5 \] \[ y = 12 \]

Thus, the realtor used \( 5 \) pillows on every twin bed and \( 12 \) pillows on every queen bed.

The realtor used 5 pillows on every twin bed and 12 pillows on every queen bed.