Questions: An investor purchased a house 8 years ago for 46,000. This year it was appraised at 69,500. Complete parts (a) through (d) below. (a) A linear equation V=mt+b, 0 ≤ t ≤ 15, represents the value V of the house for 15 years after it was purchased. Determine m and b. m= b= (Type integers or decimals.)

An investor purchased a house 8 years ago for 46,000. This year it was appraised at 69,500. Complete parts (a) through (d) below.
(a) A linear equation V=mt+b, 0 ≤ t ≤ 15, represents the value V of the house for 15 years after it was purchased. Determine m and b.
m=
b=
(Type integers or decimals.)

Transcript text: An investor purchased a house 8 years ago for $\$ 46,000$. This year it was appraised at $\$ 69,500$. Complete parts (a) through (d) below. (a) A linear equation $\mathrm{V}=\mathrm{mt}+\mathrm{b}, 0 \leq \mathrm{t} \leq 15$, represents the value V of the house for 15 years after it was purchased. Determine $m$ and $b$. \[ \begin{array}{l} \mathrm{m}=\square \\ \mathrm{b}=\square \end{array} \] (Type integers or decimals.)

Solution

Solution Steps

To determine the linear equation $ V = mt + b $ that represents the value of the house over time, we need to find the slope $ m $ and the y-intercept $ b $. The slope $ m $ can be calculated using the formula for the slope of a line, which is the change in value divided by the change in time. The y-intercept $ b $ is the initial value of the house when $ t = 0 $.

Step 1: Determine the Slope $ m $

To find the slope $ m $ of the linear equation $ V = mt + b $, we use the formula for the slope of a line:

\[ m = \frac{\text{change in value}}{\text{change in time}} = \frac{69500 - 46000}{8} = 2937.5 \]

Step 2: Determine the Y-Intercept $ b $

The y-intercept $ b $ is the initial value of the house when $ t = 0 $. Since the house was purchased for \$46,000, we have:

\[ b = 46000 \]