Problem 28 - Prealgebra

Question

You and your friend spent a total of \(\$ 15\) for lunch. Your friend's lunch cost \(\$ 3\) more than yours did. How much did you spend for lunch? \(\mathbf{F} \$ 6\) G \(\$ 7\) \(\mathbf{H} \$ 8\) J \$9

Step-by-Step Solution

Answer

You spent \( \$6 \) for lunch.

Step 1: Define Variables

Let \( x \) represent the cost of your lunch in dollars. Then, your friend's lunch cost would be \( x + 3 \).

Step 2: Write an Equation

According to the problem, the total cost of the lunches is \( \$ 15 \). Therefore, the equation is \( x + (x + 3) = 15 \).

Step 3: Solve the Equation

First, combine like terms: \( 2x + 3 = 15 \). Then, subtract 3 from both sides to get \( 2x = 12 \). Finally, divide by 2 to solve for \( x \): \( x = 6 \).

Step 4: Verify Solution

Your lunch cost \( \\(6\) and your friend's cost was \( 6 + 3 = \\)9 \). The total is \( 6 + 9 = \$15 \), which matches the problem statement.

Key Concepts

Variable Definition

In every word problem, identifying what you need to find is crucial. Here, we begin by setting up our variables. For instance, to solve how much you spent on lunch, we assign a variable, say \(x\), to represent the unknown value, the cost of your lunch. Defining variables simplifies the problem and provides a way to translate it into a mathematical equation.

• The variable \(x\) stands for your lunch cost in dollars.
• Your friend's lunch cost is expressed as \(x + 3\) since it is \(\$3\) more than yours.

By establishing these expressions for the costs, we now have a clear path to relate them to the total spent.

Equation Solving

Once we have defined our variables, the next step is to form an equation. This equation captures the essence of the problem using algebra, allowing us to solve for the unknown.
For this lunch expense problem, the equation becomes:\[ x + (x + 3) = 15 \]

• We start by expanding the expression: \( x + x + 3 \).
• Combine like terms to simplify: \( 2x + 3 \).
• To isolate \(2x\), subtract \(3\) from both sides, resulting in: \( 2x = 12 \).
• Finally, divide by \(2\) to solve for \(x\): \( x = 6 \).

The end result, \(x = 6\), indicates your lunch cost in dollars. These steps highlight the systematic process of solving equations in algebra.

Verification of Solution

After finding a solution, verifying its accuracy is vital. This ensures the answer makes sense within the context of the problem.
In our scenario, we found that:

• Your lunch cost is \( \\(6 \).
• Your friend’s lunch cost is \( \\)6 + 3 = \\(9 \).
• The total cost, therefore, is \( 6 + 9 = \\)15 \).

This matches the original problem's conditions, confirming correctness. Verification steps are crucial. They not only check the solution but also solidify understanding, preventing careless mistakes. Such a practice is an essential habit in algebra and enhances confidence in problem-solving abilities.