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 $44$ twin-size blankets and $35$ queen-size blankets, using a total of $526$ yards of fabric. Last week, the members completed $23$ twin-size blankets and $35$ queen-size blankets, which required $442$ 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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Algebra 2

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

44

twin-size blankets and

35

queen-size blankets, using a total of

526

yards of fabric. Last week, the members completed

23

twin-size blankets and

35

queen-size blankets, which required

442

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.

Denote Fabric Usage Equations: Let's denote the amount of fabric used for each twin-size blanket as $t$ and for each queen-size blanket as $q$ . We have two weeks of data, which gives us two equations. The first week's production of $44$ twin-size and $35$ queen-size blankets using $526$ yards of fabric gives us the equation $44t + 35q = 526$ .
Week $1$ Production Data: The second week's production of $23$ twin-size and $35$ queen-size blankets using $442$ yards of fabric gives us the equation $23t + 35q = 442$ .
System of Equations: We now have a system of two equations with two variables: $\newline$ $1$ . $44t + 35q = 526$ $\newline$ $2$ . $23t + 35q = 442$ $\newline$ We will use elimination to solve for $t$ and $q$ . To eliminate $q$ , we can subtract the second equation from the first.
Elimination Method: Subtracting the second equation from the first, we get: $\newline$ $(44t + 35q) - (23t + 35q) = 526 - 442$ $\newline$ This simplifies to $21t = 84$ .
Subtract Equations: Dividing both sides of the equation $21t = 84$ by $21$ , we find that $t = 4$ . This means that each twin-size blanket uses $4$ yards of fabric.
Solve for $t$ : Now that we know the value of $t$ , we can substitute it back into one of the original equations to solve for $q$ . We'll use the second equation: $23t + 35q = 442$ . Substituting $t = 4$ , we get $23(4) + 35q = 442$ .
Substitute $t$ into Equation: Calculating the first term, we have $23 \times 4 = 92$ , so the equation becomes $92 + 35q = 442$ .
Calculate for $q$ : Subtracting $92$ from both sides of the equation $92 + 35q = 442$ , we get $35q = 350$ .
Final Fabric Usage Results: Dividing both sides of the equation $35q = 350$ by $35$ , we find that $q = 10$ . This means that each queen-size blanket uses $10$ yards of fabric.
Final Fabric Usage Results: Dividing both sides of the equation $35q = 350$ by $35$ , we find that $q = 10$ . This means that each queen-size blanket uses $10$ yards of fabric.We have found that a twin-size blanket uses $4$ yards of fabric and a queen-size blanket uses $10$ yards of fabric.