Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ The members of a sewing circle are making blankets to give to shelters. This week, they made $14$ twin-size blankets and $48$ queen-size blankets, using a total of $440$ yards of fabric. Last week, the members completed $32$ twin-size blankets and $12$ queen-size blankets, which required $224$ total yards of fabric. How much fabric is used for the different sizes of blankets? $\newline$ A twin-size blanket uses $\_$ yards of fabric and a queen-size one uses $\_$ yards.

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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

The members of a sewing circle are making blankets to give to shelters. This week, they made

14

twin-size blankets and

48

queen-size blankets, using a total of

440

yards of fabric. Last week, the members completed

32

twin-size blankets and

12

queen-size blankets, which required

224

total yards of fabric. How much fabric is used for the different sizes of blankets?

\newline

A twin-size blanket uses

\_

yards of fabric and a queen-size one uses

\_

yards.

Equations Setup: Let's denote the amount of fabric used for each twin-size blanket as $t$ and for each queen-size blanket as $q$ . We are given that $14$ twin-size blankets and $48$ queen-size blankets use a total of $440$ yards of fabric, which gives us the equation $14t + 48q = 440$ .
Initial Equations: From the previous week, we know that $32$ twin-size blankets and $12$ queen-size blankets required $224$ yards of fabric, which gives us the equation $32t + 12q = 224$ .
Elimination Process: We now have a system of two equations: $\newline$ $1$ . $14t + 48q = 440$ $\newline$ $2$ . $32t + 12q = 224$ $\newline$ We will use elimination to solve this system. To eliminate one of the variables, we can multiply the second equation by $4$ to match the coefficient of $q$ in the first equation.
Variable Elimination: Multiplying the second equation by $4$ gives us $128t + 48q = 896$ . Now we have: $\newline$ $1$ . $14t + 48q = 440$ $\newline$ $2$ . $128t + 48q = 896$
Solving for $t$ : We subtract the first equation from the second equation to eliminate $q$ . This gives us $114t = 456$ .
Substitute $t$ into Equation: Dividing both sides of the equation $114t = 456$ by $114$ gives us $t = 4$ . This means that each twin-size blanket uses $4$ yards of fabric.
Solving for q: Now we substitute $t = 4$ into the first equation $14t + 48q = 440$ to find $q$ . This gives us $14(4) + 48q = 440$ , which simplifies to $56 + 48q = 440$ .
Solving for q: Now we substitute $t = 4$ into the first equation $14t + 48q = 440$ to find $q$ . This gives us $14(4) + 48q = 440$ , which simplifies to $56 + 48q = 440$ . Subtracting $56$ from both sides of the equation $56 + 48q = 440$ gives us $48q = 384$ .
Solving for q: Now we substitute $t = 4$ into the first equation $14t + 48q = 440$ to find $q$ . This gives us $14(4) + 48q = 440$ , which simplifies to $56 + 48q = 440$ . Subtracting $56$ from both sides of the equation $56 + 48q = 440$ gives us $48q = 384$ . Dividing both sides of the equation $48q = 384$ by $48$ gives us $14t + 48q = 440$ $0$ . This means that each queen-size blanket uses $14t + 48q = 440$ $1$ yards of fabric.