Jake buys a new car for $18,259. Each year [tex]$x$[/tex] after he buys the car, its value [tex]$y$[/tex] depreciates by $445. Which equation models the relationship between [tex]$x$[/tex] and [tex]$y$[/tex]?
A. [tex]$y=-445 x-18259$[/tex]
B. [tex]$y=445 x+18259$[/tex]
C. [tex]$y=-445 x+18259$[/tex]
D. [tex]$y=445 x-18259$[/tex]
Answer (1)
The car's value depreciates by $445 each year, indicating a linear relationship with a negative slope.
The initial value of the car is $18 , 259 , which serves as the y-intercept.
The equation is formed by substituting the slope and y-intercept into the linear equation y = m x + b .
The equation that models the relationship between x and y is y = − 445 x + 18259 .
Explanation
Understanding the Problem The problem states that Jake buys a car for $18 , 259 and each year the car's value depreciates by $445 . We need to find an equation that models this relationship, where x represents the number of years after the purchase and y represents the value of the car after x years.
Setting up the Equation Since the car's value decreases each year, this is a linear relationship with a negative slope. The initial value of the car is the y-intercept. The general form of a linear equation is y = m x + b , where m is the slope and b is the y-intercept. In this case, the slope m is the amount the car depreciates each year, which is − $445 , and the y-intercept b is the initial value of the car, which is $18 , 259 .
Forming the Equation Substituting the given values into the equation, we get: y = − 445 x + 18259
Final Answer Therefore, the equation that models the relationship between x and y is y = − 445 x + 18259 .
Examples
Understanding depreciation is crucial in personal finance. For example, if you buy a laptop for $1200 and it depreciates by $200 each year, after 3 years, its value would be $1200 − 3 × $200 = $600 . This concept helps in making informed decisions about when to sell or replace assets, considering their decreasing value over time. Similarly, businesses use depreciation to account for the reduction in value of their assets, which affects their financial statements and tax obligations.
