Problem 13 - Prealgebra

Question

You have saved some money. Your friend has saved \(\$ 20\) more than you. Write an expression in simplest form that represents the total amount of money you and your friend have saved.

Step-by-Step Solution

Answer

The total savings can be expressed as \( 2x + 20 \).

Step 1: Defining Variables

Let \( x \) represent the amount of money that you have saved in dollars.

Step 2: Expressing Your Friend's Savings

Since your friend has saved \( \$ 20 \) more than you, we can express your friend's savings as \( x + 20 \).

Step 3: Formulating the Total Savings Expression

To find the total amount of money you and your friend have saved, we add your savings and your friend's savings. This gives us the expression \( x + (x + 20) \).

Step 4: Simplifying the Expression

Simplify the expression \( x + (x + 20) \) by combining like terms: \( x + x + 20 = 2x + 20 \).

Key Concepts

Understanding Variables in Algebraic Expressions

Variables are a fundamental part of algebraic expressions. They act as placeholders or symbols, usually letters like \( x \), \( y \), or \( z \), that represent unknown values or quantities that can change. In the context of our example, the variable \( x \) represents the amount of money you have saved. Using variables allows you to write general expressions or equations applicable to multiple situations.

• **Variables are flexible**: They can take on different numerical values depending on the situation.
• **Naming your variables**: Choosing a variable is a flexible process – you can use any letter, but it is helpful to use a letter that has meaning, like \( x \) for savings.
• **Replacing variables**: Once you know the numerical value of a variable, you can substitute that number back into the expression to find a specific value.

This helps not only in crafting expressions that align with the problem but also in simplifying complex problems by turning them into smaller, manageable parts.

Simplifying Expressions

Once variables are established, the next step in working with algebraic expressions is simplifying them. Simplifying means to rewrite the expression in the most concise and efficient form. For the given exercise, you simplify the expression \( x + (x + 20) \) representing your total savings. The goal is to make the expression as straightforward as possible without altering its value.

• **Removing unnecessary operations**: Start by eliminating parentheses; in this example, you distribute the addition over \( x + 20 \) to obtain \( x + x + 20 \).
• **Reordering terms**: Rearrange the terms so that similar ones are next to each other.
• **Conclusion of simplification**: The original expression simplifies to \( 2x + 20 \). This step helps in more easily tracking computations or further manipulations.

By simplifying expressions, calculations become easier to manage, paving the way for clearer problem-solving and insights.

Combining Like Terms

Combining like terms is an essential process in simplifying algebraic expressions. Like terms are terms in an expression that have identical variable parts. Only these terms can be added or subtracted. In our example, combining like terms is vital to condensing the expression \( x + x + 20 \) into its simplest version.

• **Identifying like terms**: Determine which terms have the same variable raised to the same power; in \( x + x \), both are like terms since they contain the variable \( x \).
• **Adding coefficients**: Add the numerical coefficients of the like terms together. Here, the coefficients of \( x \) are both 1, so you add them: \( 1x + 1x = 2x \).
• **Resulting expression**: After combining the like terms, the expression simplifies to \( 2x + 20 \), where \( 20 \) is a constant term and does not combine with \( x \).

Understanding how to effectively combine like terms is crucial for working with complex expressions, reducing errors, and ensuring clarity in computations.