Exercise 40 Page 248

We have to look at each car separately.

Original Car

It is given that the original car has an initial value of $20000$ dollars and the value decreases by $1500$ dollars each year. If we let $x$ represent the number of years that passes and $y$ represent the value of the car, we can write an equation to describe the situation algebraically. To determine the value of the original car after $5$ years, we can substitute $x=5$ into the equation and solve for $y.$ The value of the car at year $5$ will be $12500$ dollars.

Different Car

For the different car to have the same value as the original car in $5$ years, it must also be worth $12500$ dollars. Let's use a linear equation in slope-intercept form to represent the value of the different car. In this form $b$ is the initial value and $m$ is the rate per year by which the value decreases. We will assume that $(5,12500)$ is a solution to the equation. That will ensure this car meets the requirements of the exercise. Let's begin by substituting $(5,12500)$ into the equation for $(x,y).$ Now we can arbitrarily choose the initial value of the car. Suppose this car was initially worth $18000$ dollars. Let's substitute $b=18000$ into our equation and solve for $m.$ If this car is initially worth $18000$ dollars, it must decrease by $1100$ dollars each year to reach $12500$ after $5$ years. We can now complete the equation. Let's graph both equations to ensure that both cars are worth $12500$ dollars after $5$ years.

Since both lines pass through the point $(5,12500),$ we can conclude that both cars are worth $12500$ dollars in year $5.$