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?
a) let x = number of houses with floor plan 1
let y = number of houses with floor plan 2
x + y = 56, as there are 56 total houses available.
b) 3x = y, because there are 3 times as many houses of the second floor plan available.
c) x + y = 56
3x = y
Substitute 3x for y
x + 3x = 56
4x = 56
x = 14
3x = y
42 = y
There are 4 houses of floor plan 1 available, and 32 houses of floor plan 2 available.
d) from part 1: x + y = 56
y = 56 - x
The y intercept is 56.
Set y = 0 to find the x intercept.
0 = 56 - x
x = 56
from part 2: 3x = y
The y intercept is 0, which means the x intercept is 0 as well.
If you solved by graphing, your answer would be the point (14, 42) as found by substitution in the form (x, y)
a) Which mover provides a better deal for 2 hours of work? How did you arrive at your answer?
a) The equation that illustrates the situation is:
x + y = 56
b) The equation that illustrates the situation is:
y = 3x
c) We can solve the system of equations using substitution.
Step 1: Substitute the value of y from equation b) into equation a).
x + (3x) = 56
4x = 56
Step 2: Solve for x.
4x = 56
x = 56/4
x = 14
Step 3: Substitute the value of x back into equation b) to find y.
y = 3x
y = 3(14)
y = 42
Therefore, there are 14 homes available with floor plan #1 and 42 homes available with floor plan #2.
d) The intercepts of the equation from part a) can be found by setting x or y to 0 and solving for the other variable.
When x = 0, the equation becomes:
0 + y = 56
y = 56
So, the y-intercept is (0, 56).
When y = 0, the equation becomes:
x + 0 = 56
x = 56
So, the x-intercept is (56, 0).
The equation from part b) is a linear equation, so it does not have an x or y-intercept. The graph of this equation will be a straight line passing through the origin (0, 0).
If we solve the system by graphing, the lines corresponding to the equations will intersect at the point (14, 42) since that is the solution obtained through substitution.