selling prices of homes can depend on several factors such as size and age the table shows the selling price 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 system of linear equations whose solution gives a, b, c.
(b) Solve the system
(c) Pridict the price of a home that is 10 years old and has 2500 square feet
(a) To find the values of a, b, and c, we need to set up a system of linear equations using the given data points.
Let's consider the first data point (P = 190, A = 20, S = 2):
P = a + bA + cS
190 = a + 20b + 2c
Following the same process for the other two data points, we get:
320 = a + 5b + 3c
50 = a + 40b + c
So, the system of linear equations is:
a + 20b + 2c = 190
a + 5b + 3c = 320
a + 40b + c = 50
(b) Now, let's solve the system of linear equations to find the values of a, b, and c.
We can use various methods to solve the system, such as substitution, elimination, or matrix methods. Here, we'll use the method of elimination to solve for a, b, and c:
Multiply the first equation by 2:
2a + 40b + 4c = 380
Multiply the second equation by -1:
-a - 5b - 3c = -320
Add the above two equations together:
2a + 40b + 4c + (-a) - 5b - 3c = 380 + (-320)
This simplifies to:
a + 35b + c = 60
Now we have two equations:
a + 35b + c = 60 (Equation 1)
a + 40b + c = 50 (Equation 2)
If we subtract Equation 2 from Equation 1, we get:
0a + -5b + 0c = 10
-5b = 10
b = -2
Substituting b = -2 into Equation 2 gives us:
a + 40(-2) + c = 50
a - 80 + c = 50
a + c = 130
So, we have two equations:
a + c = 130 (Equation 1)
b = -2 (Equation 2)
(c) To predict the selling price of a home that is 10 years old (A = 10) and has 2500 square feet (S = 2500), we can use the equation P = a + bA + cS, substituting the known values of a, b, and c that we've found:
P = a + bA + cS
P = 130 + (-2)(10) + c(2500)
P = 130 - 20 + 2500c
P = 110 + 2500c
Thus, the predicted price of a home that is 10 years old and has 2500 square feet is P = 110 + 2500c.