Question

Amy makes a gas stop at a station where gas is selling at \(\$ 3.15\) per gallon, including taxes. While there, she buys a bottle of water for \(\$ 1.65\). Her total is \(\$ 40.71\). If Amy bought \(g\) gallons of gas, which equation models Amy's purchase? A. \(1.64 g+3.15=40.71\) B. \(1.65 \mathrm{~g}+40.71=3.15\) C. \(3.15 g+1.65=40.71\) D. \(3.15 g+40.71=1.65\)

Short answer

The correct equation is \( 3.15 g + 1.65 = 40.71 \).

Step-by-step solution

- Identify Variables and Costs

First, identify the variables and costs given in the problem. Let the number of gallons of gas bought be represented by \( g \). The cost per gallon of gas is \( \$ 3.15 \). The cost of the bottle of water is \( \$ 1.65 \). The total amount spent is \( \$ 40.71 \).

- Model the Cost of Gas

To find the total cost of the gas, multiply the number of gallons by the cost per gallon: \( 3.15 g \).

- Account for the Bottle of Water

Add the cost of the bottle of water to the total cost of the gas. This gives us the equation: \( 3.15 g + 1.65 \).

- Set Up the Total Cost Equation

Set the sum of the gas cost and the water cost equal to the total amount spent: \( 3.15 g + 1.65 = 40.71 \).

- Compare with Given Options

Compare the derived equation with the given options and find the matching one. The correct option is: \( 3.15 g + 1.65 = 40.71 \).

Key concepts

Solving Equations

Solving equations is a fundamental concept in algebra. It involves finding the value of a variable that makes an equation true. In this exercise, we were given a real-life context where Amy bought gas and a bottle of water, and we needed to find out how to model her purchases mathematically.
To solve an equation, follow these steps:

• Identify what you need to find (the variable).
• Write down the equation that models the situation.
• Isolate the variable on one side of the equation.
• Simplify the equation by performing operations like addition, subtraction, multiplication, or division.
• Verify your solution by substituting the variable back into the original equation.

In our problem, we set up the total cost equation as follows:
Let the number of gallons of gas Amy bought be represented by the variable \( g \). The steps from the solution guide us to form the equation \( 3.15g + 1.65 = 40.71 \), which accurately represents her total purchase.

Variables in Algebra

A variable is a symbol, often a letter, that represents a number we don't know yet. In algebra, variables are used to define equations and expressions. For our example, we used \( g \) to represent the number of gallons of gas Amy bought.
Key points about variables:

• They can represent unknown values.
• Variables can be manipulated using algebraic operations.
• They help in forming equations to solve real-world problems.

In our exercise, the variable \( g \) helped us model the situation by setting up the relation between the gallons of gas, the price per gallon, and the total cost. By understanding the role of variables, you can better tackle various algebra problems and represent scenarios accurately.

Linear Equations

Linear equations are equations of the first degree, which means the variable is not raised to any power greater than one. They are written in the form \( mx + b = c \), where \( m \) and \( b \) are constants and \( x \) is the variable. In our example, we used a linear equation to determine how many gallons of gas Amy bought.
A linear equation representing our scenario is:
\( 3.15g + 1.65 = 40.71 \)

Steps to work with linear equations:

• Combine like terms, if needed.
• Isolate the variable by performing inverse operations (like subtraction, division).
• Solve for the variable to find its value.

Understanding linear equations helps in solving many everyday problems where relationships are directly proportional, like calculating costs, distances, or rates. This foundational algebra concept is crucial for tackling more complex math topics in the future.