Write a system of equations to describe the situation below, solve using elimination, 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 $5$ twin beds and $3$ queen beds and used a total of $62$ pillows. At another house, she used $34$ pillows to spruce up $5$ twin beds and $1$ 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 elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, 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

5

twin beds and

3

queen beds and used a total of

62

pillows. At another house, she used

34

pillows to spruce up

5

twin beds and

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.

Define Variables: Let $x$ be the number of pillows on every twin bed and $y$ be the number of pillows on every queen bed. We can write two equations based on the given information: $\newline$ For the first house: $5x + 3y = 62$ (Equation $1$ ) $\newline$ For the second house: $5x + 1y = 34$ (Equation $2$ ) $\newline$ We will use these equations to solve for $x$ and $y$ using the elimination method.
Elimination Method: To eliminate one of the variables, we can multiply Equation $2$ by $-3$ so that when we add it to Equation $1$ , the $y$ terms will cancel out. $\newline$ $-3(5x + 1y) = -3(34)$ $\newline$ $-15x - 3y = -102$ (Equation $3$ )
Combine Equations: Now we add Equation $3$ to Equation $1$ to eliminate $y$ : $\$5x + 3y$ + ( $-15$ x - $3$ y) = $62$ + ( $-102$ )\)This simplifies to: $5x - 15x + 3y - 3y = 62 - 102$ $-10x = -40$
Solve for x: We can now solve for $x$ by dividing both sides of the equation by $-10$ : $\newline$ $-10x / -10 = -40 / -10$ $\newline$ $x = 4$ $\newline$ So the realtor used $4$ pillows on every twin bed.
Substitute $x$ into Equation: Now that we have the value for $x$ , we can substitute it back into one of the original equations to solve for $y$ . We'll use Equation $2$ : $\newline$ $5(4) + 1y = 34$ $\newline$ $20 + y = 34$
Solve for y: Subtract $20$ from both sides of the equation to solve for y: $\newline$ $20 + y - 20 = 34 - 20$ $\newline$ $y = 14$ $\newline$ So the realtor used $14$ pillows on every queen bed.