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 $4$ twin beds and $1$ queen bed and used a total of $34$ pillows. At another house, she used $76$ pillows to spruce up $4$ twin beds and $4$ 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 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

4

twin beds and

1

queen bed and used a total of

34

pillows. At another house, she used

76

pillows to spruce up

4

twin beds and

4

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.

Equation Setup: Let's denote the number of pillows on each twin bed as $t$ and the number of pillows on each queen bed as $q$ . In the first house, the realtor used $4$ twin beds and $1$ queen bed with a total of $34$ pillows, which gives us the equation $4t + q = 34$ .
Elimination of Variable: In the second house, the realtor used $4$ twin beds and $4$ queen beds with a total of $76$ pillows, which gives us the equation $4t + 4q = 76$ .
Substitution and Solving: We now have a system of two equations. We need to eliminate one of the variables, $t$ or $q$ . We choose to eliminate $q$ because its coefficients are the same in both equations, which makes it easier to eliminate.
Final Solution: To eliminate $q$ , we subtract the first equation from the second equation. This gives us $4t + 4q - (4t + q) = 76 - 34$ , which simplifies to $3q = 42$ .
Final Solution: To eliminate $q$ , we subtract the first equation from the second equation. This gives us $4t + 4q - (4t + q) = 76 - 34$ , which simplifies to $3q = 42$ .We divide both sides of the equation $3q = 42$ by $3$ to solve for $q$ . This gives us $q = 42 / 3$ , which simplifies to $q = 14$ .
Final Solution: To eliminate $q$ , we subtract the first equation from the second equation. This gives us $4t + 4q - (4t + q) = 76 - 34$ , which simplifies to $3q = 42$ .We divide both sides of the equation $3q = 42$ by $3$ to solve for $q$ . This gives us $q = 42 / 3$ , which simplifies to $q = 14$ .We substitute $q = 14$ into the first equation $4t + q = 34$ and solve for $4t + 4q - (4t + q) = 76 - 34$ $0$ . This gives us $4t + 4q - (4t + q) = 76 - 34$ $1$ .
Final Solution: To eliminate $q$ , we subtract the first equation from the second equation. This gives us $4t + 4q - (4t + q) = 76 - 34$ , which simplifies to $3q = 42$ .We divide both sides of the equation $3q = 42$ by $3$ to solve for $q$ . This gives us $q = 42 / 3$ , which simplifies to $q = 14$ .We substitute $q = 14$ into the first equation $4t + q = 34$ and solve for $4t + 4q - (4t + q) = 76 - 34$ $0$ . This gives us $4t + 4q - (4t + q) = 76 - 34$ $1$ .We subtract $4t + 4q - (4t + q) = 76 - 34$ $2$ from both sides of the equation $4t + 4q - (4t + q) = 76 - 34$ $1$ to isolate $4t + 4q - (4t + q) = 76 - 34$ $4$ . This gives us $4t + 4q - (4t + q) = 76 - 34$ $5$ , which simplifies to $4t + 4q - (4t + q) = 76 - 34$ $6$ .
Final Solution: To eliminate $q$ , we subtract the first equation from the second equation. This gives us $4t + 4q - (4t + q) = 76 - 34$ , which simplifies to $3q = 42$ . We divide both sides of the equation $3q = 42$ by $3$ to solve for $q$ . This gives us $q = 42 / 3$ , which simplifies to $q = 14$ . We substitute $q = 14$ into the first equation $4t + q = 34$ and solve for $4t + 4q - (4t + q) = 76 - 34$ $0$ . This gives us $4t + 4q - (4t + q) = 76 - 34$ $1$ . We subtract $4t + 4q - (4t + q) = 76 - 34$ $2$ from both sides of the equation $4t + 4q - (4t + q) = 76 - 34$ $1$ to isolate $4t + 4q - (4t + q) = 76 - 34$ $4$ . This gives us $4t + 4q - (4t + q) = 76 - 34$ $5$ , which simplifies to $4t + 4q - (4t + q) = 76 - 34$ $6$ . We divide both sides of the equation $4t + 4q - (4t + q) = 76 - 34$ $6$ by $4t + 4q - (4t + q) = 76 - 34$ $8$ to solve for $4t + 4q - (4t + q) = 76 - 34$ $0$ . This gives us $3q = 42$ $0$ , which simplifies to $3q = 42$ $1$ .