1. Imagine 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 56 houses for sale with two floor plans still available. Use x to represent floor plan one and y to represent floor plan two. Write an equation that illustrates the situation. X+y=56
b. The sales representative later indicates that there are three 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. x =3y=56
c. Use the equations from Parts 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?
a.
x+y=56
is correct
b.
"there are three times as many homes available with the second floor plan than the first"
means:
number of homes with the second floor plan is greater than that of the first, in the proportion
y = 3x, or
3x - y = 0
c. the system of equations is:
x + y = 56 ... (1)
3x - y = 0 ... (2)
rewrite (2) as y=3x, and substitute y into (1) to get:
x + 3x = 56
4x = 56
x = 14, therefore y = 56-14 = 42.
d. intercepts for (1):
x+y=56
can be found by setting y=0 and solve for x, and similarly for x=0 and solve for y.
To determine the number of each type of floor plan available, we will use substitution to solve the system of equations from parts a and b.
Step 1: Let's start with the equation from part a: X + y = 56.
Step 2: Now let's solve the equation from part b to express x in terms of y: x = 3y.
Step 3: Substitute the value of x from step 2 into the equation from step 1: 3y + y = 56.
Step 4: Combine like terms: 4y = 56.
Step 5: Solve for y by dividing both sides by 4: y = 56/4 = 14.
Step 6: Substitute the value of y into the equation from step 2 to find x: x = 3 * 14 = 42.
Therefore, there are 42 houses with floor plan one (x) and 14 houses with floor plan two (y) available in the housing community.
Now let's move on to part d.
For part a, the intercepts of the equation X + y = 56 can be found by setting one of the variables to 0 and solving for the other.
- Setting x = 0, we get y = 56. So the y-intercept is (0, 56).
- Setting y = 0, we get x = 56. So the x-intercept is (56, 0).
For part b, the intercepts of the equation x = 3y = 56 can be found in the same way.
- Setting x = 0, we get 0 = 3y = 56, which has no solution since 0 is not equal to 3 times any number.
- Setting y = 0, we get x = 3 * 0 = 0. So the y-intercept is (0, 0).
To find where the lines intersect if we were to solve the system by graphing, we would plot the points (42, 14) and (0, 56) for the equation from part a, and the point (0, 0) for the equation from part b. The lines would intersect at the point of intersection, which represents the solution to the system of equations.