Math Snap

STEP 1

Assumptions1. The initial value of the house is $42,000. The value of the house8 years later is$67,5003. The value of the house changes linearly over time4. The linear equation representing the value of the house is $V=mt+b$, where $V$ is the value of the house, $m$ is the slope of the line (rate of change of the house's value), $t$ is the time in years, and $b$ is the y-intercept (initial value of the house)

STEP 2

First, we need to find the slope $m$ of the line. The slope is the rate of change of the house's value over time. It can be found by subtracting the initial value of the house from its value8 years later, and then dividing by the time difference.
$$m = \frac{V_{8\, years} - V_{initial}}{t_{8\, years} - t_{initial}}$$

STEP 3

Now, plug in the given values for the initial value of the house, its value8 years later, and the time difference to calculate the slope.
m = \frac{\($\)67,500 - \($\)42,000}{8\, years -0\, years}

STEP 4

Calculate the slope.
m = \frac{\($\)67,500 - \($\)42,000}{8} = \($\)3,187.\, per\, year

STEP 5

The y-intercept $b$ is the initial value of the house, which is given as $42,000.
b = \($\)42,000

STEP 6

Now that we have the slope $m$ and the y-intercept $b$, we can write the linear equation representing the value of the house.
$$V = mt + b$$

STEP 7

Plug in the values for $m$ and $b$ to write the equation.
V = \($\)3,187.5t + \($\)42,000

STEP 8

To estimate in how many years after purchase the house will be worth $72,500, we can set$V$to$72,500 in the equation and solve for $t$.
\($\)72,500 = \($\)3,187.5t + \($\)42,000

STEP 9

Subtract $42,000 from both sides of the equation to isolate the term with$t$.
\($\)72,500 - \($\)42,000 = \($\)3,187.5t

STEP 10

Divide both sides of the equation by $3,187.5 to solve for$t$.
t = \frac{\($\)72,500 - \($\)42,000}{\($\)3,187.5}

Calculate the value of $t$.
t = \frac{\($\)72,500 - \($\)42,000}{\($\)3,187.5} \approx9.6\, yearsSo, the house will be worth $72,500 approximately9.6 years after it was purchased.