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.
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.
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.
d) What are the intercepts of the equation from part a of this problem? What are the intercepts from part b of this problem? Where would the lines intersect if you solved the system by graphing?
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.
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.
c) Use elimination to determine how many houses with each floor plan are available. Explain how you arrived at your answer.
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?
b) Which mover provides a better deal for 15 hours of work? How did you arrive at your answer?
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.
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1.
a) Let x represent floor plan #1 and y represent floor plan #2. Since there are a total of 56 houses available, we can write the equation: x + y = 56.
b) The sales representative indicates that there are 3 times as many homes available with the second floor plan than the first. This means the number of floor plan #2 houses is three times the number of floor plan #1 houses. Using the variables x and y, we can write the equation: y = 3x.
c) To solve the system of equations using substitution, we substitute the value of y from equation (b) into equation (a): x + 3x = 56. Simplifying this equation gives us 4x = 56, and dividing both sides by 4 gives x = 14. Substituting this value back into equation (a), we find that y = 3(14) = 42. Therefore, there are 14 houses with floor plan #1 and 42 houses with floor plan #2.
d) In equation (a), the intercepts represent the number of houses with each floor plan when the other floor plan has zero houses. The intercept for floor plan #1 is (14, 0) and for floor plan #2 is (0, 42). If we were to graph the system of equations, the lines would intersect at the point (14, 42).
2.
a) Let x represent floor plan #1 and y represent floor plan #2. Since there are 38 homes available, we can write the equation: x + y = 38.
b) Floor plan #1 sells for $175,000 and floor plan #2 sells for $200,000. The total value of all available houses can be expressed by the equation: 175,000x + 200,000y = 7,200,000.
c) To solve the system of equations using elimination, we can multiply equation (a) by 175,000 to make the coefficients of x in both equations the same. This gives us 175,000x + 175,000y = 6,650,000.
Subtracting this equation from equation (b) gives us 25,000y = 550,000. Dividing both sides by 25,000 gives y = 22. Substituting this value back into equation (a), we find x + 22 = 38, which gives x = 16. Therefore, there are 16 houses with floor plan #1 and 22 houses with floor plan #2.
3.
a) Heavy Lifters charges an $80 fee plus $35 per hour, so for 2 hours of work, the cost would be 80 + (35 * 2) = $150.
Quick Move charges $55 per hour with no additional fees, so for 2 hours of work, the cost would be 55 * 2 = $110.
Therefore, Quick Move provides the better deal for 2 hours of work.
b) Heavy Lifters would charge 80 + (35 * 15) = $605 for 15 hours of work.
Quick Move would charge 55 * 15 = $825 for 15 hours of work.
Therefore, Heavy Lifters provides the better deal for 15 hours of work.
c) To find for what values of h Quick Move offers the better deal, we can set up the inequality: 55h < 80 + 35h. Simplifying this inequality gives us 20h < 80, and dividing both sides by 20 gives h < 4. Therefore, for any value of h less than 4, Quick Move offers the better deal.