Letting $$ x=1 $$ correspond to the year 2017, Darnell has determined the slope-intercept form of the equation for the line that passes through the points representing the value of his car in 2017, $$ (1,20,499) $$ and the value of his car in 2021, $$ (5,17,000) $$, to be Use this equation to, find an estimate of the value of Darnell's car in 2023. Round your answer to the nearest cent. Answer

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To estimate the value of Darnell's car in 2023, we'll first determine the slope-intercept form of the line that models the car's value over time.

**Given:**
- Year 2017 corresponds to $$ x = 1 $$ with a car value of $$ \$20,499 $$.
- Year 2021 corresponds to $$ x = 5 $$ with a car value of $$ \$17,000 $$.

**Step 1: Find the Slope (m)**
The slope $$ m $$ of the line is calculated using the formula:
$$
m = \frac{y_2 - y_1}{x_2 - x_1}
$$
$$
m = \frac{17,000 - 20,499}{5 - 1} = \frac{-3,499}{4} = -874.75
$$

**Step 2: Determine the Y-Intercept (b)**
Using the slope-intercept form $$ y = mx + b $$ and plugging in one of the points (let's use $$ (1, 20,499) $$):
$$
20,499 = (-874.75)(1) + b
$$
$$
b = 20,499 + 874.75 = 21,373.75
$$

**Step 3: Write the Equation**
$$
y = -874.75x + 21,373.75
$$

**Step 4: Estimate the Value in 2023**
First, determine the corresponding $$ x $$ value for 2023:
$$
2023 - 2017 = 6 \text{ years later} \Rightarrow x = 1 + 6 = 7
$$

Now, plug $$ x = 7 $$ into the equation:
$$
y = -874.75(7) + 21,373.75
$$
$$
y = -6,123.25 + 21,373.75 = 15,250.50
$$

**Estimated Value in 2023:**  
$$\boxed{\$15,\!250.50}$$
The estimated value of Darnell's car in 2023 is \$15,250.50.

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