Buying a Home
For most people, buying a house is a great investment that can offer security in an uncertain world, but buying a house is also a commitment.
Application Practice
Answer the following questions. Use Equation Editor to write mathematical expressions and equations. First, save this file to your hard drive by selecting Save As from the File menu. Click the white space below each question to maintain proper formatting.
1. Suppose you are in the market for a new home and are interested in a new housing community under construction in a different city.
a) The sales representative informs you that there are two floor plans still available, and that there are a total of 72 houses available. The sales representative later indicates that there are five times as many homes available with the second floor plan than the first. Write a system of equations that illustrates this situation. Use x to represent floor plan #1 and y to represent floor plan #2.
Answer Box (Type your work and answer below)
Hint: Translate the problem into two linear equations by identifying keywords that represent a mathematical operation or symbol.
b) Use the system of equations from part 1a and solve this system using the substitution method to determine how many of each type of floor plan is available. Explain the steps you used to solve the problem.
Answer Box (Type your work and answer below)
Hint: See section 8.2.
2. As you are leaving the community, you notice another new community just down the street. Because you are in the area, you decide to inquire about it.
a) The sales representative here tells you they also have two floor plans available, but they only have 50 homes available. The representative tells you that floor plan #1 sells for $150,000 and floor plan #2 sells for $100,000. She also mentions that all the available houses combined are worth $7 million. Write a system of equations that illustrates this situation. Use x to represent floor plan #1 and y to represent floor plan #2.
Answer Box (Type your work and answer below)
Hint: Translate the problem into two linear equations by identifying keywords that represent a mathematical operation or symbol.
b) Use elimination method to determine how many houses with each floor plan are available. Explain how you arrived at your answer.
Answer Box (Type your work and answer below)
Hint: See Section 8.3.
3. You recently started the paperwork to purchase your new home, and you were just notified that you can move into the house in 2 weeks. You decide to hire a moving company, but are unsure which company to choose. You search online and are interested in contacting two companies, Heavy Lifters and Quick Move, to discuss their rates. Heavy Lifting charges a $130 fee plus $25 per hour. Quick Move charges a $55 fee plus $50 per hour.
For what values h (hours) does Quick Move offer the better deal? Express your answer as an inequality. Explain how you reached your answer.
Answer Box (Type your work and answer below)
Hint: Graph the linear functions Q(x) and H(x) to have a visual of which company offers the better deal. The better deal depends on the point of intersection of the lines, which defines when both deals are the same. Use an online graphic calculator to do and post your graph to explain and support your answer. An alternative is to set up a linear inequality that represents Quick Move offers a better deal. Solve for the variable of the linear inequality you set up.
If you expect to pass this course by having others do your homework and tests for you, think again.
Please note the correct spelling of the subject.
1a) The problem states that there are two floor plans available - plan #1 and plan #2. Let's use the variables x and y to represent the number of houses available for each floor plan.
According to the problem, there are a total of 72 houses available. So, the first equation would be:
x + y = 72
The second piece of information says that there are five times as many homes available with the second floor plan than the first floor plan. So, the second equation would be:
y = 5x
b) Now let's solve the system of equations using the substitution method.
From the second equation y = 5x, we can substitute this value into the first equation:
x + 5x = 72
6x = 72
x = 12
Now substitute this value of x back into one of the original equations. Using the second equation:
y = 5(12)
y = 60
So, there are 12 houses available with floor plan #1 and 60 houses available with floor plan #2.
2a) In this scenario, let's use the variables x and y to represent the number of houses available for each floor plan, where x represents floor plan #1 and y represents floor plan #2.
The problem states that there are a total of 50 houses available. So, the first equation would be:
x + y = 50
The problem also mentions that floor plan #1 sells for $150,000 and floor plan #2 sells for $100,000. The total worth of all available houses is $7 million. So, the second equation would be:
150,000x + 100,000y = 7,000,000
b) Let's use the elimination method to solve this system of equations.
Multiply the first equation by 100,000 to match the coefficients of y in both equations:
100,000(x + y) = 100,000(50)
100,000x + 100,000y = 5,000,000
We can then subtract this equation from the second equation to eliminate y:
(150,000x + 100,000y) - (100,000x + 100,000y) = 7,000,000 - 5,000,000
50,000x = 2,000,000
x = 40
Now substitute the value of x back into one of the original equations, such as the first equation:
40 + y = 50
y = 10
So, there are 40 houses available with floor plan #1 and 10 houses available with floor plan #2.
3. Let's use the variables x and y to represent the number of hours for Heavy Lifters and Quick Move, respectively.
According to the problem, Heavy Lifters charges a $130 fee plus $25 per hour, so the cost function for Heavy Lifters would be:
H(x) = 25x + 130
Quick Move charges a $55 fee plus $50 per hour, so the cost function for Quick Move would be:
Q(x) = 50x + 55
We want to find for what values of x (hours) Quick Move offers the better deal. This means we want to find when the cost function for Quick Move is less than the cost function for Heavy Lifters:
Q(x) < H(x)
Substituting the cost functions, we have:
50x + 55 < 25x + 130
Simplifying the inequality:
25x < 75
x < 3
Therefore, Quick Move offers a better deal for any number of hours less than 3.