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Algebra is like a toolbox for solving real-world problems, and at the heart of algebra are algebraic expressions. An algebraic expression combines numbers, variables (like the letter x), and operation symbols such as addition, subtraction, multiplication, and division. In the fuel cost problem, we deal with a variable x, representing the uncertain number of miles Nola will drive. To depict mathematical relationships, we construct expressions like x / 42, which shows the number of gallons Nola needs - miles driven divided by the car's efficiency.
To understand how to build these expressions, start with identifying the variable and constants. Here, the variable is the number of miles (x), and the constant is the car's gas mileage (42 miles per gallon). The division operation tells us how many units of one quantity (gallons of gas) are needed per unit of another (miles driven). Once you grasp these components and their interactions, you'll be equipped to translate real-world scenarios into algebraic expressions with ease. Always remember, algebraic expressions are the foundation upon which equations are built, allowing us to solve for unknowns and make predictions.

Understanding unit rates is essential in daily life, especially when comparing different items or calculating costs. A unit rate describes how many units of the first type of quantity correspond to one unit of the second type of quantity. In our fuel cost problem, we're given a unit rate of 42 miles per gallon. This tells us Nola's car can travel 42 miles on a single gallon of gas.
Applying this concept, we create a practical calculation to estimate expenses or resources. For Nola's trip, we need to find out how much gas she'll buy for a certain number of miles. We apply the unit rate by dividing the total miles by miles per gallon, yielding the total gallons needed—a direct application of the unit rate. Consequently, this understanding enables students to compare costs, optimize budgets, and make informed decisions based on efficiency—all from mastering the concept of the unit rate.

When dealing with expenses that change depending on certain conditions—like the number of miles driven—we're engaging in variable cost calculations. It's a vital skill for budgeting and forecasting in various situations. In Nola's case, the cost of gas is a variable cost as it changes with the distance she travels.
The algebraic expression for Nola's total gas expense, which we determined as x / 42 * 2.38, multiplicatively combines the variable quantity (x miles) with the unit rate (cost per gallon). This expression gives us a framework to compute the total cost dynamically, as the variable x changes. To calculate specific costs, we simply plug in the number of miles for x and perform the arithmetic. The ability to calculate variable costs is important in personal finance, business, and economics, as it helps in estimating how changing conditions affect overall expenses.
In education, fostering an intuitive understanding of variable cost is equally important, as it prepares students not only for academic achievements but also for real-world financial literacy and decision-making.