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Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.\newlineA 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 33 twin beds and 55 queen beds and used a total of 7575 pillows. At another house, she used 1212 pillows to spruce up 11 queen bed. How many decorative pillows did the realtor arrange on each bed?\newlineThe realtor used ___\_\_\_ pillows on every twin bed and ___\_\_\_ pillows on every queen bed.

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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.\newlineA 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 33 twin beds and 55 queen beds and used a total of 7575 pillows. At another house, she used 1212 pillows to spruce up 11 queen bed. How many decorative pillows did the realtor arrange on each bed?\newlineThe realtor used ___\_\_\_ pillows on every twin bed and ___\_\_\_ pillows on every queen bed.
  1. First equation formulation: First equation based on 33 twin beds and 55 queen beds using 7575 pillows: 3t+5q=753t + 5q = 75.
  2. Second equation formulation: Second equation based on 11 queen bed using 1212 pillows: 0t+1q=120t + 1q = 12.
  3. Augmented matrix creation: Write the system of equations as an augmented matrix: ∣3amp;5amp;75 0amp;1amp;12∣\left| \begin{array}{cc|c} 3 & 5 & 75 \ 0 & 1 & 12 \end{array} \right|
  4. Matrix row operations: Use row operations to solve the matrix. Since the second row is already solved for qq (0t+1q=120t + 1q = 12), we can use it to find the value of qq.
  5. Substitute and solve for qq: q=12q = 12. Substitute qq into the first equation: 3t+5(12)=753t + 5(12) = 75.
  6. Calculate twin bed pillows: Calculate the number of pillows on each twin bed: 3t+60=753t + 60 = 75.
  7. Solve for tt: Subtract 6060 from both sides: 3t=153t = 15.
  8. Final solution: Divide by 33 to find tt: t=153t = \frac{15}{3}.
  9. Final solution: Divide by 33 to find tt: t=153t = \frac{15}{3}. t=5t = 5. So, the realtor uses 55 pillows on every twin bed and 1212 pillows on every queen bed.

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