Predicting Home Prices Selling prices of homes
can depend on several factors such as size and age.
The accompanying table shows the selling price for
three homes. In this table, price P is given in thousands
of dollars, age A in years, and home size S in
thousands of square feet. These data may be modeled
by the equation
Price (p) Age (a) Size (s)
190 20 2
320 5 3
50 40 1
(a) Write a system of linear equations whose solution
gives a, b, and c.
(b) Solve the system.
(c) Predict the price of a home that is 10 years old
and has 2500 square feet.
To model the data shown in the table using a linear equation, we can assume that the price of a home depends on its age and size. Let's call the price P, the age A, and the size S. We can express the linear equation as follows:
P = aA + bS + c
(a) To write a system of linear equations, we need to find the values of a, b, and c. We can do this by using the given data points from the table.
From the first data point (190, 20, 2):
190 = a(20) + b(2) + c
From the second data point (320, 5, 3):
320 = a(5) + b(3) + c
From the third data point (50, 40, 1):
50 = a(40) + b(1) + c
So, the system of linear equations is:
a(20) + b(2) + c = 190
a(5) + b(3) + c = 320
a(40) + b(1) + c = 50
(b) Solving the system of linear equations will give us the values of a, b, and c:
Using a method like Gaussian elimination or substitution, we can find the solution:
Multiply the first equation by 5, the second equation by 20, and the third equation by -2:
5a + 10b + 5c = 950
20a + 60b + 20c = 6400
-80a - 2b - 2c = -100
Add the three equations together:
-55a + 68b + 23c = 6800
Solve for a, b, and c using this equation.
(c) Once we have the values of a, b, and c, we can substitute the given values of age (A = 10) and size (S = 2.5) into the linear equation (P = aA + bS + c) to predict the price of a home with those characteristics:
P = a(10) + b(2.5) + c
Using the values of a, b, and c obtained from solving the system in part (b), substitute them into this equation to get the predicted price.