To solve this problem, we need to set up and solve a system of equations based on the information provided.
Let's define:
- [tex]\( d_1 \)[/tex] as the number of doors in a one-story house.
- [tex]\( d_2 \)[/tex] 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:
[tex]\[ 6d_1 + 6d_2 = 144 \][/tex]
For the east side:
[tex]\[ 8d_1 + 6d_2 = 162 \][/tex]
We now have the system of equations:
[tex]\[ \begin{cases}
6d_1 + 6d_2 = 144 \\
8d_1 + 6d_2 = 162
\end{cases} \][/tex]
To solve this system, we can use the method of elimination.
First, we subtract the first equation from the second equation to eliminate [tex]\( d_2 \)[/tex]:
[tex]\[ (8d_1 + 6d_2) - (6d_1 + 6d_2) = 162 - 144 \][/tex]
This simplifies to:
[tex]\[ 2d_1 = 18 \][/tex]
Solving for [tex]\( d_1 \)[/tex]:
[tex]\[ d_1 = 9 \][/tex]
Now we substitute [tex]\( d_1 = 9 \)[/tex] back into the first equation to solve for [tex]\( d_2 \)[/tex]:
[tex]\[ 6(9) + 6d_2 = 144 \][/tex]
[tex]\[ 54 + 6d_2 = 144 \][/tex]
[tex]\[ 6d_2 = 90 \][/tex]
[tex]\[ d_2 = 15 \][/tex]
Therefore, each one-story house has [tex]\( 9 \)[/tex] doors, and each two-story house has [tex]\( 15 \)[/tex] doors.
So, filling in the blanks:
Each one-story house has 9 doors, and each two-story house has 15 doors.
