Question

3. The value of a car in relation to the number of miles driven is given by the linear

function y = -9x + 55000.
a. Interpret the meaning of the rate of change and initial value for the linear function.

Answer 1

General form of a linear equation:
`y=mx+b`

(where m is the slope and b is the y-intercept)

A linear equation has a constant rate of change which is represented by the slope (m). It is the rate at which one quantity is changing with respect to another quantity.

Question 3

Part (a)

Given equation:
`y=-9x+55000`

where:

• y = value of car (in dollars)
• x = number of miles driven

The rate of change (slope) is -9 and represents that for every 1 mile driven, the value of the car decreases by $9.

The initial value is the y-intercept (b) as this is the value when x = 0.

Therefore, the initial value of car is $55,000.

Part (b)

To find the number of miles at which the car be valued at $35,002, set the equation to $35,002 and solve for x:

`\implies -9x+55000=35002`

`\implies -9x=35002-55000`

`\implies -9x=-19998`

`\implies x=(-19998)/(-9)`

`\implies x=2222`

Therefore, the car will be valued at $35,002 after 2,222 miles.

Question 4

Given equation:
`y=0.39x+2.55`

where:

• y = price of carrots (in dollars)
• x = weight (in oz)

The rate of change (slope) is 0.39 and represents that for every increase of 1 oz of weight in carrots, the price of the carrots increases by $0.39

The initial value is the y-intercept (b) as this is the value when x = 0.

Therefore, the initial cost of the carrots is $2.55.

Part (b)

To find the weight of carrots you can buy for $6.06, set the equation to $6.06 and solve for x:

`\implies 0.39x+2.55=6.06`

`\implies 0.39x=6.06-2.55`

`\implies 0.39x=3.51`

`\implies x=(3.51)/(0.39)`

`\implies x=9`

Therefore, you can buy 9 oz of carrots for $6.06

Part (c)

To find out how much it costs to buy 21 oz of carrots, substitute x = 21 into the equation and solve for y:

`\implies y=0.39(21)+2.55`

`\implies y=10.74`

Therefore, it costs $10.74 to buy 21 oz of carrots.