I keep confusing myself when I look at this word problem. I am not sure if I am setting it up correctly. Any advice would be appreciated. It has several parts to it.
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.
would this be 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.
would this be x+3y=56?
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.
Now here is where I get confused looking at it I believe the answer is x=14 and y=42 I am just not sure how to set it up in an algebra equation to solve with the substitution method.
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?
would x and y intercept eachother at 56?
Your equation for the second part b) is not correct.
It said "there are 3 times as many homes available with the second floor plan than the first"
which translates into y = 3x
subbing that back into the first gives you
x + (3x) = 56
x = 14 and then y = 3(14) = 56
the way you had it:
x+y=56 and
x+3y=56 you have stated conflicting conditions on x and y, and the only solution would be if y=0 and x = 56.
(three times zero is still zero)
for d) in x+y=56, if x=0, then y=56
if y=0 then x=56
so the intercepts are (56,0) and (0,56)
your second equation only has one intercept, namely the origin (0,0)
Sure enough, they would intersect at (14,42)
Thank you so much for your help! That makes more sense. I am getting ready to have a test next week so I am trying to practice as much as I can with these word problems. I do much better if the problem is right there staring me in the face but if I have to turn it into a problem I get stuck.
Take care!
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
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.
175,000x + 200,000y = 7,2000,000
c) Use elimination to determine how many houses with each floor plan are available. Explain how you arrived at your answer.
a) Yes, you are correct. The equation for this situation is x + y = 56, where x represents the number of houses with floor plan #1 and y represents the number of houses with floor plan #2.
b) You are almost correct. Since the sales representative said there are 3 times as many homes available with the second floor plan than the first, the equation would be x + 3y = 56.
c) To solve the system of equations using substitution, follow these steps:
1. From equation a), solve for x: x = 56 - y
2. Substitute this value of x into equation b): 56 - y + 3y = 56
3. Simplify the equation: 56 + 2y = 56
4. Subtract 56 from both sides: 2y = 0
5. Divide both sides by 2: y = 0
6. Substitute this value of y back into x = 56 - y: x = 56 - 0
Therefore, the solution to the system of equations is x = 56 and y = 0, meaning there are 56 houses with floor plan #1 and 0 houses with floor plan #2.
d) In part a), the intercepts of the equation x + y = 56 would occur when either x or y is equal to 0.
- When x = 0, the equation becomes 0 + y = 56, so the y-intercept is (0, 56).
- When y = 0, the equation becomes x + 0 = 56, so the x-intercept is (56, 0).
In part b), the intercepts of the equation x + 3y = 56 would also occur when x or y is equal to 0.
- When x = 0, the equation becomes 0 + 3y = 56, so the y-intercept is (0, 18.67).
- When y = 0, the equation becomes x + 0 = 56, so the x-intercept is (56, 0).
If you were to graph these equations, the lines would intersect at (56, 0), as this point satisfies both equations.