Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. 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? Each one-story house has _____ doors, and each two-story house has _____ doors.

Question

Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.  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?  Each one-story house has _____ doors, and each two-story house has _____ doors.

Bytelearn · Accepted Answer

Define Variables: Define the variables for the number of doors in each type of house. Let x be the number of doors in one one-story house. Let y be the number of doors in one two-story house.Write Equations: Write the system of equations based on the given information. For the west side of the street: 8 one-story houses and 7 two-story houses require a total of 168 doors. 8x + 7y = 168  For the east side of the street: 1 two-story house requires a total of 16 doors. 0x + 1y = 16Augmented Matrix: Write the augmented matrix for the system of equations. The augmented matrix is: [8 7 | 168] [0 1 | 16]Row Operations: Use row operations to find the reduced row echelon form of the matrix. 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. We multiply the second row by -7 and add it to the first row: -7 * [0 1 | 16] = [0 -7 | -112] Adding this to the first row: [8 7 | 168] + [0 -7 | -112] = [8 0 | 56] The updated matrix is now: [8 0 | 56] [0 1 | 16]Solve for x: Solve for x by dividing the first row by 8. [8 0 | 56] becomes [1 0 | 7] after dividing the entire row by 8. 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.

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