Question

Write a system of equations to describe the situation below, solve using elimination, 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 5 one-story houses and 5 two-story houses on the west side of the street, which require a total of 115 doors. On the east side, there are 2 one-story houses and 5 two-story houses, which require a total of 85 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?

Answer 1

"There are 5 one-story houses and 5 two-story houses on the west side of the street, which require a total of 115 doors." translates to 5x + 5y = 115

"there are 2 one-story houses and 5 two-story houses, which require a total of 85 doors." translates to 2x + 5y = 85

`\begin{gathered} \text{Solve by elimination} \\ 5x+5y=115 \\ 2x+5y=85 \\ \text{Multiply the second equation by -1, then add the equations together.} \\ (5x+5y=115) \\ -1(2x+5y=85 \\ \downarrow \\ 5x+5y=115 \\ -2x-5y=-85 \\ \text{Add these equations to eliminate y:} \\ 3x=30 \\ \text{Divide both sides by 3} \\ (3x)/(3)=(30)/(3) \\ x=10 \\ \text{Substitute }x=10\text{ to any of the equation} \\ 5x+5y=115 \\ 5(10)+5y=115 \\ 50+5y=115 \\ 5y=115-50 \\ 5y=65 \\ (5y)/(5)=(65)/(5) \\ y=13 \end{gathered}`

Substitute x = 10 for one story houses, and y = 13 for two story houses.

`\begin{gathered} \text{West Side Houses} \\ 5x+5y=115 \\ 5(10)=50 \\ 5(13)=65 \\ \text{There are 50 doors for one-story house and 65 for the two-story house on west side} \\ \\ \text{East Side houses} \\ 2x+5y=85 \\ 2(10)=20 \\ 5(13)=65 \\ \text{There are 20 doors for one-story house and 65 for the two-story house on the east side} \end{gathered}`