Problem 35 - Prealgebra

Question

Write an expression in simplest form that represents the total amount in situation. You bought 5 folders that each cost \(x\) dollars, a calculator for \(\$ 45,\) and a set of pens for \(\$ 3 .\)

Step-by-Step Solution

Answer

The simplest form of the expression is \(5x + 48\).

Step 1: Identify the Cost of Items

First, identify the cost of each item purchased. You bought 5 folders, with each folder costing \(x\) dollars, so the total cost for folders is \(5x\) dollars. In addition, the calculator costs \(\\(45\) and the set of pens costs \(\\)3\).

Step 2: Set up the Expression

Next, set up an algebraic expression to reflect the total cost. Add the cost of the folders, the calculator, and the set of pens: \(5x + 45 + 3\).

Step 3: Simplify the Expression

Combine the constant terms \(45\) and \(3\) to simplify the expression. Adding these gives \(45 + 3 = 48\). Therefore, the simplified expression is \(5x + 48\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves combining like terms and reducing expressions to their simplest form. This process makes calculations easier and expressions more understandable. In the exercise, we are given the expression for calculating the total cost of several purchases. Simplifying it means combining the constant numbers.
Let's look at the expression:

• Start with the expression: \(5x + 45 + 3 \)
• Combine constants: Add \(45\) and \(3\), which are the fixed costs since they don't have the variable \(x\).
• The result of adding them is \(48\).

Thus, the simplified expression becomes \(5x + 48\), which combines all the constant costs into one term and keeps the variable component separate for easy understanding of its dependency on \(x\).
Remember, simplifying is about making the expression leaner without changing its value, so each simplification step must be done with care.

Variables in Expressions

Variables in algebra serve as placeholders for unknown values and provide flexibility in calculations. In our example, the variable \(x\) represents the cost of one folder. Since we bought multiple folders, \(5x\) represents the total cost related to the variable \(x\).
Here is how variables work within expressions:

• Variables can represent any value; this allows for expressions to be generalized.
• In "\(5x\)", \(x\) is the unknown cost per folder, while \(5\) indicates how many folders were bought.

Understanding how variables fit into expressions helps decode how changes to one part of the purchase affect the total cost. If the cost of each folder changes, the total change is simply captured by modifying \(x\). Variables therefore enable a dynamic representation of a problem, giving both flexibility and precision.

Cost Calculation

Cost calculation involves aggregating the cost of different items to find a total expense. By setting up an expression in our problem, we effectively organized this calculation in a structured way. Here’s the breakdown:

• The expression combines the purchase costs of separate items to provide a total cost.
• The fixed costs (calculator and pen set) are added as constants, while the variable cost (folders) is represented by \(5x\).
• By evaluating the expression with a specific value of \(x\), the exact total cost can be calculated easily.

Cost calculation with expressions is efficient as it groups different cost items together and allows for quick adjustments. When prices change, just substitute the new value of \(x\), and the expression will compute a new total. This approach is not just useful for solving problems, but is widely used in budgeting and financial forecasting as well.