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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.

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.

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?

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1 Answer

a. We know that there are exactly 56 houses and each house has one of the two floor plans, so x + y = 56

b. As stated simply, y = 3x : if you multiply the number of houses with floor plan 1 by 3, you get the number of houses with floor plan 2.

c. Our equations are x+y=56 and y=3x. Substituting y from the second equation (3x in other words) into the first equation, we get x + 3x = 56. Combining like terms: 4x = 56. Solving for x: x = 14. There are three times as many y's as there x's, so y = 3x = 3*14 = 42. You can verify that these answers satisfy both equations simultaneously.

d. The first equation was x + y = 56. The intercepts occur where either x or y is zero. Making x zero, y= 56. Making y zero, x = 56. So the intercepts are at (56, 0) and (0, 56). The second equation was 3x = y. Making x zero, y = 0. Making y zero, x = 0. So it crosses the origin (0, 0). The lines would intersect at the point that satisfies both equations, which have figured out is (14, 42).