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?

And your question is???

Answer

a) Let's assume that there are x houses available with floor plan #1 and y houses available with floor plan #2. Since there are a total of 56 houses available, we can write the equation:

x + y = 56

b) According to the sales representative, there are 3 times as many homes available with the second floor plan than the first. This can be expressed with the equation:

y = 3x

c) To solve the system of equations using substitution, we can substitute the value of y from equation b) into equation a).

From equation b), we have y = 3x. We can substitute this into equation a):

x + (3x) = 56

Simplify the equation:

4x = 56

Divide both sides by 4:

x = 14

Now, substitute the value of x back into equation b):

y = 3(14)

y = 42

So, there are 14 houses available with floor plan #1 and 42 houses available with floor plan #2.

d) Let's find the intercepts of the equations.

For part a), the equation is x + y = 56. To find the x-intercept, we set y = 0 and solve for x:

x + 0 = 56
x = 56

So, the x-intercept is (56, 0). To find the y-intercept, we set x = 0 and solve for y:

0 + y = 56
y = 56

So, the y-intercept is (0, 56).

For part b), the equation is y = 3x. To find the x-intercept, we set y = 0 and solve for x:

0 = 3x
x = 0

So, the x-intercept is (0, 0). To find the y-intercept, we set x = 0 and solve for y:

y = 3(0)
y = 0

So, the y-intercept is (0, 0).

If we graph the equations, the lines would intersect at the point (14, 42), which represents the solution to the system.