Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ 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 $3$ twin beds and $5$ queen beds and used a total of $75$ pillows. At another house, she used $12$ pillows to spruce up $1$ queen bed. How many decorative pillows did the realtor arrange on each bed? $\newline$ The realtor used $\_\_\_$ pillows on every twin bed and $\_\_\_$ pillows on every queen bed.

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Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.

\newline

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

3

twin beds and

5

queen beds and used a total of

75

pillows. At another house, she used

12

pillows to spruce up

1

queen bed. How many decorative pillows did the realtor arrange on each bed?

\newline

The realtor used

\_\_\_

pillows on every twin bed and

\_\_\_

pillows on every queen bed.

First equation formulation: First equation based on $3$ twin beds and $5$ queen beds using $75$ pillows: $3t + 5q = 75$ .
Second equation formulation: Second equation based on $1$ queen bed using $12$ pillows: $0t + 1q = 12$ .
Augmented matrix creation: Write the system of equations as an augmented matrix: $\left| \begin{array}{cc|c} 3 & 5 & 75 \ 0 & 1 & 12 \end{array} \right|$
Matrix row operations: Use row operations to solve the matrix. Since the second row is already solved for $q$ ( $0t + 1q = 12$ ), we can use it to find the value of $q$ .
Substitute and solve for $q$ : $q = 12$ . Substitute $q$ into the first equation: $3t + 5(12) = 75$ .
Calculate twin bed pillows: Calculate the number of pillows on each twin bed: $3t + 60 = 75$ .
Solve for $t$ : Subtract $60$ from both sides: $3t = 15$ .
Final solution: Divide by $3$ to find $t$ : $t = \frac{15}{3}$ .
Final solution: Divide by $3$ to find $t$ : $t = \frac{15}{3}$ . $t = 5$ . So, the realtor uses $5$ pillows on every twin bed and $12$ pillows on every queen bed.