Problem 79 - Algebra A Combined Function

Question

Write each algebraic expression described. Write an expression with 4 terms that simplifies to \(3 x-4\)

Step-by-Step Solution

Answer

One possible expression is \(2x + x - 2 - 2 + 3x = 3x - 4\).

Step 1: Understand the Problem

We need to write an algebraic expression that has four terms and simplifies to \(3x - 4\). This means after simplifying or combining like terms, the final expression should result in the combination of \(3x\) and \(-4\).

Step 2: Choose Terms for the Expression

A useful strategy is to start with \(3x\) and \(-4\) as part of the expression and then add two more terms that do not affect the final outcome when simplified. For example, selecting terms like \(2x\) and \(-2x\), or \(5\) and \(-5\), can help as they will cancel each other out.

Step 3: Construct the Initial Expression

Let's construct the expression: \(2x + x - 2 - 2 + 3x\). At this stage, you can see we have used four terms: \(2x\), \(x\), \(-2\), and \(-2\), plus an extra \(3x\) because we want it to combine with others to create a four-term expression before simplification.

Step 4: Simplify the Expression

Combine like terms: Start with combining \(2x\) and \(x\) and \(3x\) which altogether become \(6x\), then combine the constants \(-2\) and \(-2\) to get \(-4\).\[2x + x + 3x - 2 - 2 = 6x - 4 = 3x - 4\]Simplifying results in \(3x - 4\).

Step 5: Verify the Simplification

Check that simplifying each step correctly involves combining \(2x + x + 3x = 6x\), and \(-2 - 2 = -4\), ultimately resulting in the expression \(6x - 4\), which correctly simplifies to \(3x - 4\).

Key Concepts

Simplifying Expressions

Simplifying an expression means making it as straightforward as possible. Think of it as cleaning up your room. First, identify parts that can be combined together to create a neater version. In algebra, this involves performing operations that shorten the expression, while retaining its value.

To start simplifying, look for opportunities to combine terms. This could mean adding together similar elements, like variables with the same letter, or constant numbers. Remember, the goal is to express the same idea using less clutter, resulting in a clearer and more concise expression.

• Reduce the number of terms
• Combine similar items
• Preserve original meaning

Simplifying makes solving algebra problems easier by reducing complexity and potential for errors.

Like Terms

In algebra, recognizing like terms is a major step in simplifying expressions. Like terms are simply terms that have the same variables raised to the same powers. For instance, in the expression \(2x + 3 + 4x - 5\), the like terms are \(2x\) and \(4x\) because they both contain the variable \(x\).

To efficiently manage an equation, it's critical to first identify these like terms. Once identified, like terms can be combined to simplify the overall expression. This is similar to how you would group similar items in your backpack - it makes it easier to see what you have and manage it effectively.

• Identify terms with the same variables and powers
• Combine them for simplicity
• Recognize constants as another set of like terms

Understanding and using like terms is a fundamental skill that eases the simplification process.

Combining Terms

Combining terms is the process of adding or subtracting like terms to simplify an expression. Once you've identified like terms, the next step is to combine them by performing the necessary arithmetic.

For instance, suppose you have \(3x + 2x - 5\). First, combine the like terms, \(3x\) and \(2x\), resulting in \(5x\). The constant term, \(-5\), remains unchanged. The expression is now simplified to \(5x - 5\), making it much more manageable.

• Add or subtract coefficients of like variables
• Simplify the constants separately
• Ensure each step maintains the expression's integrity

Combining terms not only simplifies the expression but also clarifies the problem by revealing its essential components.