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 78 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.
b. The sales representative later indicates that there are twice 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 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 = 78.
b. Y = 2X.
c. Eq1: X + Y = 78.
Eq2: Y = 2X.
In Eq1,substitute 2X for Y:
X + 2X = 78,
3X = 78,
X = 26 Homes with the 1st floor plan.
In Eq2, substitute 26 for X:
Y = 2*26 = 52 Homes with the 2nd floor plan.
d. X + Y = 78.
When X = 0,
Y = 78 = Y-Intercept.
When Y = 0,
X = 78 = X-Intercept.
Y = mx + b, The slope int. form.
Y = 2X + 0.
The Y-Int. = 0.
When Y = 0,
X = 0 = X-Int.
Solution set:(X,Y)= (26,52).
The solution set is the point where the
lines intersect.
a. To represent the situation where there are 78 houses for sale with two floor plans available, we can write the equation:
x + y = 78
Here, x represents the number of houses with floor plan one, and y represents the number of houses with floor plan two.
b. The sales representative indicated that there are twice as many homes available with the second floor plan than the first. We can write the equation to illustrate this situation as:
y = 2x
Here, y represents the number of houses with floor plan two, and x represents the number of houses with floor plan one.
c. To determine how many of each type of floor plan is available, we can solve the system of equations from parts a. and b. using the substitution method.
First, we can rearrange the equation from part b. to solve for y in terms of x:
y = 2x
Next, we substitute this expression for y into the equation from part a.:
x + (2x) = 78
3x = 78
x = 26
Now that we have the value of x, we can substitute it back into the equation y = 2x to find y:
y = 2(26)
y = 52
Therefore, there are 26 houses with floor plan one and 52 houses with floor plan two.
d. In Part a. of the problem, the equation x + y = 78 represents the situation. To find the intercepts, we can set x or y equal to zero and solve for the other variable.
When x = 0, we have y = 78, which means the y-intercept is 78.
When y = 0, we have x = 78, which means the x-intercept is 78.
In Part b. of the problem, the equation y = 2x represents the situation. To find the intercepts, we can set x or y equal to zero and solve for the other variable.
When x = 0, we have y = 0, which means the y-intercept is 0.
When y = 0, we have 0 = 2x, which means there is no x-intercept (since the equation implies that x must be 0 as well).
To determine where the lines would intersect if we solved the system by graphing, we can graph both equations on the same coordinate plane. The point at which the lines intersect is the solution to the system.
However, since we have already solved the system using the substitution method in part c., we know that the lines intersect at the point (26, 52).