Write a system of equations to describe the situation below, solve using any method, 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 $2$ queen beds and used a total of $34$ pillows. At another house, she used $46$ pillows to spruce up $5$ twin beds and $2$ queen beds. 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 any method, 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

2

queen beds and used a total of

34

pillows. At another house, she used

46

pillows to spruce up

5

twin beds and

2

queen beds. 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.

Define variables: Let's define the variables. Let $x$ be the number of pillows on each twin bed and $y$ be the number of pillows on each queen bed.
Write first equation: Based on the information given for the first house, we can write the first equation as $3x + 2y = 34$ , where $3$ is the number of twin beds and $2$ is the number of queen beds.
Write second equation: For the second house, the information given allows us to write the second equation as $5x + 2y = 46$ , where $5$ is the number of twin beds and $2$ is the number of queen beds.
Solve using elimination: We now have a system of equations: $\newline$ $1$ ) $3x + 2y = 34$ $\newline$ $2$ ) $5x + 2y = 46$ $\newline$ We can solve this system using the method of elimination or substitution. Let's use elimination to solve for $x$ and $y$ .
Eliminate y: To eliminate y, we can subtract the first equation from the second equation: $\newline$ $(5x + 2y) - (3x + 2y) = 46 - 34$ $\newline$ $5x + 2y - 3x - 2y = 12$ $\newline$ $2x = 12$ $\newline$ $x = \frac{12}{2}$ $\newline$ $x = 6$
Substitute $x$ : Now that we have the value for $x$ , we can substitute it back into one of the original equations to solve for $y$ . Let's use the first equation: $\newline$ $3x + 2y = 34$ $\newline$ $3(6) + 2y = 34$ $\newline$ $18 + 2y = 34$ $\newline$ $2y = 34 - 18$ $\newline$ $2y = 16$ $\newline$ $y = \frac{16}{2}$ $\newline$ $y = 8$