Question

There are two gas stations near your home. Station A costs $2 to drive there and back, and it sells gas for $2.50/gallon. Station B costs $5 to drive there and back, but it sells gas for $2.25/gallon. a) Let A(x) be the total cost to buy x gallons of gas from station A. Find a linear equation for A(x). b) Let B(x) be the total cost to buy x gallons of gas from station B. Find a linear equation for B(x). c) What amount of gas yields the same total cost from both stations?

Answer 1

a. The linear equation is A(x) = 2.50 x + 2

b. The linear equation is B(x) = 2.25 x + 5

c. The amount of gas yields the same total cost from both stations is 12 gallons

Explanation:

The form of the linear function is f(x) = m x + b, where

• m is the rate of change
• b is the initial value

Let us solve the question

Station A:

∵ Station A costs $2 to drive there and back

- That means the initial amount is 2

∴ b = 2

∵ it sells gas for $2.5/gallon

- That means the price per gallon is $2.50 ⇒ rate of change

∴ m = 2.50

∵ A(x) is the total cost to buy x gallons of gas

- Substitute them in the form of the linear function above

∴ A(x) = 2.50 x + 2

a. The linear equation is A(x) = 2.50 x + 2

Station B:

∵ Station B costs $5 to drive there and back

- That means the initial amount is 5

∴ b = 5

∵ it sells gas for $2.25/gallon

- That means the price per gallon is $2.25 ⇒ rate of change

∴ m = 2.25

∵ B(x) is the total cost to buy x gallons of gas

- Substitute them in the form of the linear function above

∴ B(x) = 2.25 x + 5

b. The linear equation is B(x) = 2.25 x + 5

∵ The costs of the two stations are equal

- Equate A(x) and B(x)

∴ A(x) = B(x)

∴ 2.50 x + 2 = 2.25 x + 5

- Subtract 2.25 x from both sides

∴ 0.25 x + 2 = 5

- Subtract 2 from both sides

∴ 0.25 x = 3

- Divide both sides by 0.25

∴ x = 12

c. The amount of gas yields the same total cost from both stations is 12 gallons