A home purchased in 1996 for $\$ 181,262$ was appraised at $\$ 266,694$ in 2000 . Assuming the rate of increase in the value of the home is constant, complete parts (a) through (c).
a. Write an equation for the value of the home as a function of the number of years, $x$, after 1996 .
b. Assuming that the equation in part (a) remained accurate, write an inequality that gives the range of years (until the end of 2010) when the value of the home was greater than $\$ 277,373$
c. Does it seem reasonable that this model remained accurate until the end of 2010 ?
a. Write a linear equation that describes the value of the home.
$y=\$$ (Do not include the $\$$ symbol in your answer.)

Step 1 ：The problem is asking for a linear equation that describes the value of the home. The value of the home in 1996 is the y-intercept (b) and the slope (m) is the rate of increase per year. We can find the slope by subtracting the initial value from the final value and dividing by the number of years. The equation will be in the form y = mx + b.

Step 2 ：Given: initial_value = $181,262, final_value = $266,694, initial_year = 1996, final_year = 2000.

Step 3 ：Calculate the slope (m) using the formula: \(m = \frac{final\_value - initial\_value}{final\_year - initial\_year}\).

Step 4 ：Substitute the given values into the formula: \(m = \frac{266694 - 181262}{2000 - 1996} = 21358.0\).

Step 5 ：The slope of the line, which represents the rate of increase in the value of the home per year, is $21358.

Step 6 ：The y-intercept (b), which represents the initial value of the home in 1996, is $181262.

Step 7 ：Therefore, the equation that describes the value of the home is \(y = 21358x + 181262\).

Step 8 ：\(\boxed{y = 21358x + 181262}\) is the final answer.