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 (2)
The initial value of the car is $18259, which is the y-intercept.
The car depreciates by $445 each year, which is the slope of the line.
Since the car depreciates, the slope is negative, so the slope is $-445.
The equation of the line is y = − 445 x + 18259 .
y = − 445 x + 18259
Explanation
Problem Analysis Let's analyze the problem. We are given that Jake buys a new car for $18,259. The value of the car depreciates by 445 e a c h ye a r . W e n ee d t o f in d an e q u a t i o n t ha t m o d e l s t h ere l a t i o n s hi p b e tw ee n t h e n u mb ero f ye a rs x a f t er h e b u ys t h ec a r an d t h ec a r ′ s v a l u e y$.
Identifying Slope and Intercept The initial value of the car is 18 , 259. T hi s i s t h e v a l u eo f y w h e n x=0$, so it is the y-intercept of the equation. The car depreciates by $445 each year. This means that the value of the car decreases by 445 f ore a c h ye a r t ha tp a sses . T hi s i s t h es l o p eo f t h ee q u a t i o n . S in ce t h ec a r d e p rec ia t es , t h es l o p e i s n e g a t i v e . T h ere f ore , t h es l o p e i s -445.
Forming the Equation The equation of a line is given by y = m x + b , where m is the slope and b is the y-intercept. In this case, m = − 445 and b = 18259 . Substituting these 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 instance, when buying a car, its value decreases over time due to wear and tear and market factors. The equation y = − 445 x + 18259 helps predict the car's value after a certain number of years. This knowledge assists in making informed decisions about when to sell or trade-in the car, optimizing financial planning.
The relationship between the years after Jake buys his car and its value is modeled by the equation y = − 445 x + 18259 . This indicates that the car depreciates by $445 each year from its initial value of $18,259. The correct choice based on the options given is A.
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