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 $48$ twin-size blankets and $38$ queen-size blankets, using a total of $372$ yards of fabric. Last week, the members completed $48$ twin-size blankets and $24$ queen-size blankets, which required $288$ 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

48

twin-size blankets and

38

queen-size blankets, using a total of

372

yards of fabric. Last week, the members completed

48

twin-size blankets and

24

queen-size blankets, which required

288

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: Let's denote the amount of fabric used for a twin-size blanket as $x$ yards and for a queen-size blanket as $y$ yards. We can write two equations based on the information given for this week and last week.
Formulate Equations: This week's equation: $48x + 38y = 372$ (since $48$ twin-size and $38$ queen-size blankets used $372$ yards of fabric).
Elimination Method: Last week's equation: $48x + 24y = 288$ (since $48$ twin-size and $24$ queen-size blankets used $288$ yards of fabric).
Simplify Equation: To solve the system using elimination, we can subtract the second equation from the first to eliminate $x$ . $(48x + 38y) - (48x + 24y) = 372 - 288$
Solve for y: This simplifies to: $\newline$ $48x + 38y - 48x - 24y = 372 - 288$ $\newline$ $14y = 84$
Substitute y Value: Now, divide both sides by $14$ to find the value of y: $\newline$ $\frac{14y}{14} = \frac{84}{14}$ $\newline$ $y = 6$
Solve for x: Now that we have the value of $y$ , we can substitute it back into one of the original equations to find $x$ . Let's use the second equation: $\newline$ $48x + 24(6) = 288$
Final Results: Substitute the value of $y$ into the equation: $48x + 144 = 288$
Final Results: Substitute the value of $y$ into the equation: $\newline$ $48x + 144 = 288$ Subtract $144$ from both sides to solve for $x$ : $\newline$ $48x = 288 - 144$ $\newline$ $48x = 144$
Final Results: Substitute the value of $y$ into the equation: $\newline$ $48x + 144 = 288$ Subtract $144$ from both sides to solve for $x$ : $\newline$ $48x = 288 - 144$ $\newline$ $48x = 144$ Now, divide both sides by $48$ to find the value of $x$ : $\newline$ $\frac{48x}{48} = \frac{144}{48}$ $\newline$ $x = 3$
Final Results: Substitute the value of $y$ into the equation: $\newline$ $48x + 144 = 288$ Subtract $144$ from both sides to solve for $x$ : $\newline$ $48x = 288 - 144$ $\newline$ $48x = 144$ Now, divide both sides by $48$ to find the value of $x$ : $\newline$ $\frac{48x}{48} = \frac{144}{48}$ $\newline$ $x = 3$ We have found that a twin-size blanket uses $48x + 144 = 288$ $0$ yards of fabric and a queen-size one uses $48x + 144 = 288$ $1$ yards.