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 $27$ queen-size blankets, using a total of $284$ yards of fabric. Last week, the members completed $47$ twin-size blankets and $45$ queen-size blankets, which required $497$ 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.

Full solution

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

27

queen-size blankets, using a total of

284

yards of fabric. Last week, the members completed

47

twin-size blankets and

45

queen-size blankets, which required

497

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.

Define variables: Define variables for the fabric used per blanket type. Let $t$ be the yards of fabric per twin-size blanket and $q$ be the yards of fabric per queen-size blanket. From the problem, we have the equations: $\newline$ - $14t + 27q = 284$ (this week's fabric usage) $\newline$ - $47t + 45q = 497$ (last week's fabric usage)
Use elimination: Use elimination to solve the system. Multiply the first equation by $45$ (the coefficient of $q$ in the second equation) and the second equation by $27$ (the coefficient of $q$ in the first equation): $\newline$ - $630t + 1215q = 12780$ $\newline$ - $1269t + 1215q = 13419$
Subtract equations: Subtract the first new equation from the second new equation to eliminate $q$ : $- 1269t + 1215q - (630t + 1215q) = 13419 - 12780$ $- 639t = 639$
Solve for t: Solve for t: $\newline$ - $t = \frac{639}{639}$ $\newline$ - $t = 1$
Substitute t: Substitute $t = 1$ back into the first original equation to solve for $q$ : $\newline$ - $14(1) + 27q = 284$ $\newline$ - $14 + 27q = 284$ $\newline$ - $27q = 270$ $\newline$ - $q = 270 / 27$ $\newline$ - $q = 10$