Answered

Write a system of equations to describe the situation below, solve using any method, 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 6 one-story houses and 6 two-story houses on the west side of the
street, which require a total of 144 doors. On the east side, there are 8 one-story houses and
6 two-story houses, which require a total of 162 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.

Verified Answer :

GinnyAnswer

To solve this problem, we need to set up and solve a system of equations based on the information provided.

Let's define:
- \(\( d_1 \)\) as the number of doors in a one-story house.
- \(\( d_2 \)\) as the number of doors in a two-story house.

We are given the following information:

1. On the west side of the street, there are 6 one-story houses and 6 two-story houses, and the total number of doors required is 144.
2. On the east side of the street, there are 8 one-story houses and 6 two-story houses, and the total number of doors required is 162.

From this information, we can set up the following system of equations:

For the west side:
\(\[ 6d_1 + 6d_2 = 144 \]\)

For the east side:
\(\[ 8d_1 + 6d_2 = 162 \]\)

We now have the system of equations:
\(\[ \begin{cases} 6d_1 + 6d_2 = 144 \\ 8d_1 + 6d_2 = 162 \end{cases} \]\)

To solve this system, we can use the method of elimination.

First, we subtract the first equation from the second equation to eliminate \(\( d_2 \)\):

\(\[ (8d_1 + 6d_2) - (6d_1 + 6d_2) = 162 - 144 \]\)

This simplifies to:
\(\[ 2d_1 = 18 \]\)

Solving for \(\( d_1 \)\):
\(\[ d_1 = 9 \]\)

Now we substitute \(\( d_1 = 9 \)\) back into the first equation to solve for \(\( d_2 \)\):

\(\[ 6(9) + 6d_2 = 144 \]\)
\(\[ 54 + 6d_2 = 144 \]\)
\(\[ 6d_2 = 90 \]\)
\(\[ d_2 = 15 \]\)

Therefore, each one-story house has \(\( 9 \)\) doors, and each two-story house has \(\( 15 \)\) doors.

So, filling in the blanks:
Each one-story house has 9 doors, and each two-story house has 15 doors.

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