Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ Britney is a costume designer for the local children's theater company. Yesterday, she sewed $5$ female costumes and $1$ male costume, which used $45$ meters of fabric. Today, she sewed $1$ male costume, which used a total of $5$ meters. How many meters of fabric does each type of costume require? $\newline$ Each female costume requires $\_\_\_\_$ meters of fabric, and every male costume requires $\_\_\_\_$ meters.

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Grade 8

Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.

\newline

Britney is a costume designer for the local children's theater company. Yesterday, she sewed

5

female costumes and

1

male costume, which used

45

meters of fabric. Today, she sewed

1

male costume, which used a total of

5

meters. How many meters of fabric does each type of costume require?

\newline

Each female costume requires

\_\_\_\_

meters of fabric, and every male costume requires

\_\_\_\_

meters.

Set Equations: Let $x$ be the meters of fabric for each female costume and $y$ for each male costume. $\newline$ We get two equations: $\newline$ $5x + y = 45$ (from the first day) $\newline$ $x + y = 5$ (from the second day)
Augmented Matrix: Write the system of equations as an augmented matrix: $\begin{pmatrix} 5 & 1 \ 1 & 1 \end{pmatrix}\begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 45 \ 5 \end{pmatrix}$
Row Operations: To solve the matrix, we'll use row operations to get it into reduced row-echelon form. $\newline$ First, we'll make the element in the first row, first column a $1$ by dividing the entire first row by $5$ . $\newline$ $\begin{align*} &\begin{pmatrix} 1 & 0.2 \ 1 & 1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 9 \ 5 \end{pmatrix} \end{align*}$
Subtract Rows: Next, subtract the first row from the second row to make the element in the second row, first column a $0$ . $\newline$ $\begin{align*} &\begin{pmatrix} 1 & 0.2 \ 0 & 0.8 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 9 \ -4 \end{pmatrix} \end{align*}$
Divide Rows: Now, divide the second row by $0.8$ to make the element in the second row, second column a $1$ . $\newline$ $\newline$ \begin{align*} $\newline$ \left| \begin{array}{cc} $\newline$ $1$ & $0$ . $2$ (\newline\) $0$ & $1$ $\newline$ \end{array} \right| $\newline$ \left| \begin{array}{c} $\newline$ x (\newline\)y $\newline$ \end{array} \right| = \left| \begin{array}{c} $\newline$ $9$ (\newline\) $-5$ $\newline$ \end{array} \right| $\newline$ \end{align*} $\newline$