Scenario #1: You handle internal accounting for Anderson Homes, a land/housing developer. The company purchased land in Central Arkansas and developed 200 acres. Two orders were recently placed with a nursery by the company’s landscapers. The invoices from the orders do not list the per-item price.
a) After contacting Nursery A, they inform you that the first order that was placed was for 13 Hydrangea bushes and 4 Dogwood trees at a cost of $487. Use x to represent bushes and y to represent the trees. Write an equation that illustrates the situation.
b) Nursery A later indicates that the second order called for 6 Hydrangea bushes and 2 Dogwood trees at a cost of $232. Write an equation that illustrates this situation. Write an equation that illustrates this situation using the same variables you used in part a.
c) Use the equations from part a and b as a system of equations. Use elimination to determine the cost of each. Describe the steps you used to solve the problem.
Scenario #2: The mortgage department of the company is selling two model homes that are located on the same block. The square footage, as well as the type of model, determines the cost of these homes.
e) The second home has 1000 square feet less than twice the square feet of the first home. Write an equation that illustrates this situation using x and y to denote home #1 and home #2, respectively.
f) Together, both homes have 4,400 square feet. Write an equation that illustrates this situation using the same variables you used in part e.
g) Use substitution to determine the square footage of each house. Explain how you arrived at your answer.
h) The market gives a cost per square footage at $49.70. How much will each home cost before upgrades/amenities? Explain
Scenario #3: Your company must use a transportation service to shuttle their corporate partners from the airport for a quarterly meeting. Crespo Transportation Service charges $32 initially plus $8 per mile for every mile traveled. Siera Transportation Service charges $24 initially plus $10 per mile for every mile traveled.
i) Write an equation in two variables (x & y) that illustrates the costs for the Crespo Transportation Service.
j) Write an equation in two variables (x & y) that illustrates the costs for the Siera Transportation Service.
k) If the trip is 3 miles, how much does each one charge? If the trip is 6 miles, how much does each one charge? Explain
l) At what mileage are both services charging the same rate? Explain
m) If you were to graph this system of linear equations, what is happening “graphically”, where the services are charging same rate? Explain
I will be happy to critique your thinking on this. Please repost with that.
a) The equation representing the cost of the first order from Nursery A can be written as: 13x + 4y = 487, where x represents the cost of each Hydrangea bush and y represents the cost of each Dogwood tree.
b) The equation representing the cost of the second order from Nursery A can be written as: 6x + 2y = 232, using the same variables x and y as in part a.
c) To solve this system of equations using elimination, multiply the equations by suitable constants such that the coefficients of x or y in one equation become equal to each other. In this case, we can multiply the first equation by 6 and the second equation by 13 to make the coefficients of x in both equations equal:
Equation 1: 78x + 24y = 2922
Equation 2: 78x + 26y = 3016
Now subtract Equation 1 from Equation 2 to eliminate x:
(78x + 26y) - (78x + 24y) = 3016 - 2922
2y = 94
y = 47
Substitute the value of y into either equation (easier if we use the second equation) to find the value of x:
6x + 2(47) = 232
6x + 94 = 232
6x = 138
x = 23
Therefore, the cost of each Hydrangea bush (x) is $23 and the cost of each Dogwood tree (y) is $47.
e) The equation representing the square footage of the second home (y) in relation to the square footage of the first home (x) can be written as: y = 2x - 1000, where x represents the square footage of home #1 and y represents the square footage of home #2.
f) The equation representing the total square footage of both homes can be written as: x + y = 4400, using the same variables x and y as in part e.
g) To solve this system of equations using substitution, isolate one variable in one equation and substitute it into the other equation. In this case, we can isolate y in the equation from part e:
y = 2x - 1000
Now substitute this expression for y into the equation from part f:
x + (2x - 1000) = 4400
3x - 1000 = 4400
3x = 5400
x = 1800
Substitute the value of x into the expression for y to find the value of y:
y = 2(1800) - 1000
y = 2600
Therefore, the first home has a square footage of 1800 and the second home has a square footage of 2600.
h) To find the cost of each home before upgrades/amenities, multiply the square footage of each home by the cost per square footage of $49.70:
Cost of home #1 = 1800 * $49.70 = $89,460
Cost of home #2 = 2600 * $49.70 = $129,220
Therefore, home #1 will cost $89,460 and home #2 will cost $129,220 before upgrades/amenities.
i) The equation representing the cost for the Crespo Transportation Service can be written as: y = 8x + 32, where x represents the number of miles traveled and y represents the total cost.
j) The equation representing the cost for the Siera Transportation Service can be written as: y = 10x + 24, where x represents the number of miles traveled and y represents the total cost.
k) If the trip is 3 miles, substitute x = 3 into the equations from parts i and j:
For Crespo Transportation Service: y = 8(3) + 32 = 24 + 32 = $56
For Siera Transportation Service: y = 10(3) + 24 = 30 + 24 = $54
If the trip is 6 miles, substitute x = 6 into the equations:
For Crespo Transportation Service: y = 8(6) + 32 = 48 + 32 = $80
For Siera Transportation Service: y = 10(6) + 24 = 60 + 24 = $84
l) Set the equations from parts i and j equal to each other to find the mileage at which both services charge the same rate:
8x + 32 = 10x + 24
-2x = -8
x = 4
At 4 miles, both services charge the same rate.
m) If you were to graph this system of linear equations, the point of intersection between the two lines would be where the services are charging the same rate. The x-coordinate of this point represents the mileage at which both services charge the same rate.