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
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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 24 houses available. Use x to represent floor plan #1 and y to represent floor plan #2. Write an equation that illustrates the situation. (2 pts.)
b) The sales representative later indicates that there are 2 times as many homes available with the second floor plan than the first. Write an equation that illustrates this situation. Use the same variables you used in part a. (2 pts.)
c) Use the equations from part a and b of this exercise as a system of equations. Use substitution to determine how many of each type of floor plan is available. Describe the steps you used to solve the problem. (2 pts.)
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 12 homes available. Write an equation that illustrates the situation. Use x and y to denote floor plan #1 and floor plan #2 respectively. (3 pts.)
b) The representative tells you that floor plan #1 sells for $300,000 and floor plan #2 sells for $200,000. She also mentions that all the available houses combined are worth $3,300,000. Write an equation that illustrates this situation. Use the same variables you used in part a. (3 pts.)
c) Use elimination to determine how many houses with each floor plan are available. Explain how you arrived at your answer. (3 pts.)
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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 56 houses available. Use x to represent floor plan #1 and y to represent floor plan #2. Write an equation that illustrates the situation.
x + y = 56
b) The sales representative later indicates that there are 3 times as many homes available with the second floor plan than the first. Write an equation that illustrates this situation. Use the same variables you used in part a.
y = 3x
c) Use the equations from part a and b of this exercise as a system of equations. Use substitution to determine how many of each type of floor plan is available. Describe the steps you used to solve the problem.
x + 3x = 56 Substitute equation b into a 4x = 56
x = 56
x = 14
y = 3(14)
y=42
14 houses with plan 1, and 42 houses with plan 2.
m
y – 56 –x
56 – x = 0
-x = - 56
x = 56 (x intercept)
Part b: y=3x.
Intercepts from a: (0,56) and (56,0)
intercepts from b: are (0,0)
If graphed the two lines would intersect at (14,42)
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 38 homes available. Write an equation that illustrates the situation. Use x and y to denote floor plan #1 and floor plan #2 respectively.
x + y =38; y = (38-x)
b) The representative tells you that floor plan #1 sells for $175,000 and floor plan #2 sells for $200,000. She also mentions that all the available houses combined are worth $7,200,000. Write an equation that illustrates this situation. Use the same variables you used in part a.
175000x + 200000y= 7200000
c) Use elimination to determine how many houses with each floor plan are available. Explain how you arrived at your answer.
175000x + 200000 (38-x) = 7200000
175000x + 7600000 – 200000x = 7200000
175000x – 200000x = 7200000 – 7600000
25000x = 400000
x = 16
38-16 = 22
16 houses for $ 175, 000
22 houses for $ 200.000
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 an $80 fee plus $35 per hour. Quick Move charges $55 per hour with no additional fees.
a) Which mover provides a better deal for 2 hours of work? How did you arrive at your answer?
HL 2 (35) + 80 = 150
QM 2 (55) = 110 Better deal!
b) Which mover provides a better deal for 15 hours of work? How did you arrive at your answer?
HL 15 (35) +80 = 605 better deal!
QM 15 ( 55) = 825
c) For what values h (hours) does Quick Move offer the better deal? Express your answer as an inequality. Explain how you reached your answer.
55h < 35 h +80
55h – 35 h < 80
20h < 80
H= 80
H < 4 hr
1.
a) Let x represent floor plan #1 and y represent floor plan #2. Since there are two floor plans still available and a total of 24 houses available, we can write the equation:
x + y = 24
b) The sales representative informs that there are 2 times as many homes available with the second floor plan than the first. We can express this relationship using the equation:
y = 2x
c) To solve this system of equations, we can substitute the value of y from equation b) into equation a). Substituting y = 2x into x + y = 24, we get:
x + 2x = 24
3x = 24
x = 8
Substituting x = 8 into equation b), we find:
y = 2(8) = 16
Therefore, there are 8 houses with floor plan #1 and 16 houses with floor plan #2 available in the new housing community.
2.
a) Let x represent floor plan #1 and y represent floor plan #2. Since there are 12 homes available in the new community, we can write the equation:
x + y = 12
b) The sales representative mentions that floor plan #1 sells for $300,000 and floor plan #2 sells for $200,000. If we let x represent the number of floor plan #1 homes and y represent the number of floor plan #2 homes, the combined value of all the available houses can be expressed as:
300,000x + 200,000y = 3,300,000
c) To determine the number of houses with each floor plan, we can use the elimination method. Multiply equation a) by 200,000 to make the coefficients of y in both equations equal. This results in:
200,000x + 200,000y = 2,400,000
Subtracting this equation from equation b) (300,000x + 200,000y = 3,300,000 - (200,000x + 200,000y = 2,400,000)), we get:
100,000x = 900,000
x = 9
Substituting x = 9 into equation a), we find:
9 + y = 12
y = 3
Therefore, there are 9 houses with floor plan #1 and 3 houses with floor plan #2 available in the new community.