A new car worth \$45,000 is depreciating in value by \$5000 per year. a. Write a formula that models the car's value, y, in dollars, after x years. b. Use the formula from part (a) to determine after how many years the car's value will be \$10,000. c. Graph the formula from part (a) in the first quadrant of a rectangular coordinate system. Then show your solution to part (b) on the graph.

Verified step by step guidance

1. Step 1: To model the car's value, we recognize that the car's value decreases linearly over time. The initial value of the car is \$45,000, and it depreciates by \$5,000 per year. The formula for the car's value, y, after x years can be written as a linear equation: $y\; =\; 45000\; -\; 5000x$.
2. Step 2: To determine after how many years the car's value will be \$10,000, substitute y = 10,000 into the formula from part (a). This gives the equation: $10000\; =\; 45000\; -\; 5000x$. Solve for x by isolating the variable.
3. Step 3: Rearrange the equation from Step 2 to isolate x. Subtract 45,000 from both sides: $10000\; -\; 45000\; =\; -5000x$. Simplify the left-hand side to get: $-35000\; =\; -5000x$. Then divide both sides by -5000 to solve for x.
4. Step 4: For part (c), graph the formula $y\; =\; 45000\; -\; 5000x$ in the first quadrant of a rectangular coordinate system. The y-intercept is 45,000 (when x = 0), and the slope is -5,000, indicating the car's value decreases by \$5,000 per year. Plot the line using these points and extend it until the value reaches \$10,000.
5. Step 5: On the graph, locate the point where the line intersects y = 10,000. This x-coordinate represents the number of years it takes for the car's value to depreciate to \$10,000. Verify that this x-coordinate matches the solution from part (b).

Key Concepts

Linear Functions

A linear function is a mathematical expression that describes a relationship between two variables, typically in the form y = mx + b, where m is the slope and b is the y-intercept. In this context, the car's value decreases linearly over time, making it essential to understand how to formulate and interpret linear equations to model the depreciation.

Depreciation

Depreciation refers to the reduction in the value of an asset over time, often due to wear and tear or obsolescence. In this scenario, the car's value decreases by a fixed amount each year, which is a key aspect of understanding how to calculate future values based on a constant rate of depreciation.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate system to visualize the relationship between the variables. For this problem, it is important to graph the car's value over time to see how it changes and to identify specific values, such as when the car's worth reaches $10,000, which can be represented as an intersection point on the graph.