Problem 10 - Algebra 1

Question

You started with some money in your pocket. All you spent was \(\$ 4.65\) on lunch. You ended up with \(\$ 7.39\) in your pocket. Write an equation to find how much money you started with.

Step-by-Step Solution

Answer

The starting amount of money was \$12.04.

Step 1: Understand the problem

The problem talks about starting with an unknown amount of money, spending a certain amount on lunch, and ending with a specific amount. This can be modeled with an equation.

Step 2: Formulate the equation

Let \(x\) represent the money at the start. It is clear that spending \$4.65 from the initial amount and still having \$7.39 left can be expressed with this equation: \(x - 4.65 = 7.39\).

Step 3: Solve the equation

Next, solve for \(x\), the starting amount, by adding \$4.65 to each side of the equation. Doing so, the equation is: \(x = 7.39 + 4.65\).

Step 4: Final Calculation

Now, calculate \(x = 7.39 + 4.65\), which yields: \(x = 12.04\). So the starting amount was \$12.04.

Key Concepts

Algebraic Expressions

Algebraic expressions are the backbone of algebra. They represent quantities without fixed values, known as variables. Think of them as a recipe for baking a cake, where the ingredients can vary, but the process remains the same. In an algebraic expression, numbers, variables, and mathematical operations combine to describe a particular computation or relationship.

For instance, the expression \( x - 4.65 \) from the exercise is an algebraic expression. The variable \(x\) represents the unknown quantity we're trying to find — in this case, the amount of money you started with. The number 4.65 is a constant that symbolizes the money spent on lunch.

Creating these expressions requires an understanding of how mathematical operations interact with variables. Once you have these foundational skills, algebra becomes a powerful tool for representing and solving real-world problems.

Equation Modeling

Equation modeling is a crucial aspect of mathematics that involves representing situations numerically to find unknown quantities. It's like having a scale where you balance weights — what you do on one side, you must replicate on the other to maintain equality. Modeling a real-world scenario with an equation allows us to unlock numbers and values that we might not immediately see or measure.

In our exercise, we modeled a scenario of spending and remaining balance with the equation \( x - 4.65 = 7.39 \). This equation is a model of the financial transaction from the problem statement. When creating an equation model, it's essential to maintain balance. If you spend money, that means you take it away from what you have, which is why we subtract 4.65 from \(x\). Understanding the reason behind each part of the equation makes the math more than just manipulation of numbers — it turns it into a storytelling device.

Basic Algebra

Basic algebra revolves around finding unknowns in equations. It's the mathematical equivalent of solving a mystery. The detective work in algebra involves operations like addition, subtraction, multiplication, and division. It takes the rules of arithmetic and applies them to a broader set of problems where some information is unknown or variable.

In the case of our textbook exercise, we're using the basic algebraic operation of addition to find the original amount of money. After setting up the equation, we use the principle that adding the same amount to both sides of an equation will keep the equation balanced, which is beautifully illustrated in the solution steps. By adding 4.65 to both sides of \( x - 4.65 = 7.39 \), we isolate \(x\) and find its value, exposing the unknown just like revealing the culprit in a detective story.