P. 620 #56. 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 P=a+bA+cS

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 solve this problem, we need to first write a system of linear equations using the given data.

(a) Write a system of linear equations whose solution gives a, b, and c.

We are given three data points with the selling price (P), age (A), and home size (S) values:

Data point 1: P = 190 (price), A = 20 (age), S = 2 (size)
Data point 2: P = 320 (price), A = 5 (age), S = 3 (size)
Data point 3: P = 50 (price), A = 40 (age), S = 1 (size)

Using the equation P = a + bA + cS, we can substitute the values from each data point to form three equations:

Equation 1: 190 = a + 20b + 2c
Equation 2: 320 = a + 5b + 3c
Equation 3: 50 = a + 40b + c

These three equations form the system of linear equations whose solution will give us the values of a, b, and c.

(b) Solve the system.

To solve the system of linear equations, we can use various methods such as substitution, elimination, or matrix operations. Let's use the substitution method in this case.

From Equation 1, isolate 'a': a = 190 - 20b - 2c
Substitute this expression for 'a' into Equations 2 and 3:

320 = (190 - 20b - 2c) + 5b + 3c
50 = (190 - 20b - 2c) + 40b + c

Simplifying the equations:

130 = -15b + c (Equation 4)
-140 = 21b - c (Equation 5)

We now have a system of two equations (Equations 4 and 5) with two variables (b and c).

Adding Equations 4 and 5 eliminates the variable 'c':

-10 = 6b
b = -10/6
b = -5/3

Substitute this value for 'b' in Equation 4 or 5 to solve for 'c':

130 = -15(-5/3) + c
130 = 25/3 + c
390/3 - 25/3 = c
365/3 = c

Now, substitute the values of 'b' and 'c' into Equation 1 to solve for 'a':

190 = a + 20(-5/3) + 2(365/3)
190 = a - 100/3 + 730/3
190 = a + 630/3
190 = a + 210
190 - 210 = a
a = -20

So the solution to the system is:
a = -20
b = -5/3
c = 365/3

(c) Predict the price of a home that is 10 years old and has 2500 square feet.

To predict the price of a home, substitute the values into the equation P = a + bA + cS:

P = -20 + (-5/3)(10) + (365/3)(2500/1000)
P = -20 - 50/3 + 365/3
P = -60/3 + 365/3
P = 305/3
P ≈ 101.67

Therefore, the predicted price of a home that is 10 years old and has 2500 square feet is approximately $101,670.