Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ An employee at a construction company is ordering interior doors for some new houses that are being built. There are $8$ one-story houses and $7$ two-story houses on the west side of the street, which require a total of $168$ doors. On the east side, there are $1$ two-story house, which require a total of $16$ doors. Assuming that the floor plans for the one-story houses are identical and so are the two-story houses, how many doors does each type of house have? $\newline$ Each one-story house has $\_\_\_\_$ doors, and each two-story house has $\_\_\_\_$ doors.

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Algebra 1

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 an augmented matrix, and fill in the blanks.

\newline

An employee at a construction company is ordering interior doors for some new houses that are being built. There are

8

one-story houses and

7

two-story houses on the west side of the street, which require a total of

168

doors. On the east side, there are

1

two-story house, which require a total of

16

doors. Assuming that the floor plans for the one-story houses are identical and so are the two-story houses, how many doors does each type of house have?

\newline

Each one-story house has

\_\_\_\_

doors, and each two-story house has

\_\_\_\_

doors.

Define Variables: Define the variables for the number of doors in each type of house. $\newline$ Let $x$ be the number of doors in one one-story house. $\newline$ Let $y$ be the number of doors in one two-story house.
Write Equations: Write the system of equations based on the given information. $\newline$ For the west side of the street: $\newline$ $8$ one-story houses and $7$ two-story houses require a total of $168$ doors. $\newline$ $8x + 7y = 168$ $\newline$ For the east side of the street: $\newline$ $1$ two-story house requires a total of $16$ doors. $\newline$ $0x + 1y = 16$
Augmented Matrix: Write the augmented matrix for the system of equations. $\newline$ The augmented matrix is: $\newline$ $\begin{bmatrix} 8 & 7 & | & 168 \ 0 & 1 & | & 16 \end{bmatrix}$
Row Operations: Use row operations to find the reduced row echelon form of the matrix. $\newline$ Since the second row already has a leading $1$ in the second column, we can use it to eliminate the $y$ -term from the first row. $\newline$ We multiply the second row by $-7$ and add it to the first row: $\newline$ $-7 \times [0 1 | 16] = [0 -7 | -112]$ $\newline$ Adding this to the first row: $\newline$ $[8 7 | 168] + [0 -7 | -112] = [8 0 | 56]$ $\newline$ The updated matrix is now: $\newline$ $[8 0 | 56]$ $\newline$ $[0 1 | 16]$
Solve for $x$ : Solve for $x$ by dividing the first row by $8$ . $\newline$ $[8 0 | 56]$ becomes $[1 0 | 7]$ after dividing the entire row by $8$ . $\newline$ So, $x = 7$ .
Solve for $y$ : The second row already indicates that $y = 16$ .
Interpret Results: Interpret the results. Each one-story house has $7$ doors, and each two-story house has $16$ doors.