Problem 47 - Algebra A Combined Function

Question

Perform each indicated operation. Subtract \(4 x\) from \((7 x-3)\)

Step-by-Step Solution

Answer

The result is \(3x - 3\).

Step 1: Understand the Problem

We are given an expression \((7x - 3)\) and need to subtract \(4x\) from it. This means we will write it as \((7x - 3) - 4x\).

Step 2: Rearrange the Expression

Express the operation explicitly: \((7x - 3) - 4x = 7x - 3 - 4x\). Ensure all terms are visible and ready for simplification.

Step 3: Combine Like Terms

Look for like terms in the expression. Here, \(7x\) and \(-4x\) are like terms. Combine them: \(7x - 4x = 3x\).

Step 4: Simplify the Expression

The expression now becomes \(3x - 3\) after combining \(7x - 4x = 3x\). You cannot simplify further unless more like terms or specific values are given.

Key Concepts

Subtraction of Expressions

When dealing with algebraic expressions, subtraction involves taking one expression away from another. In the exercise given, we need to subtract \(4x\) from the expression \((7x - 3)\). This is written as \((7x - 3) - 4x\). It is important to understand that subtraction in expressions works similarly to numeric subtraction, except you are dealing with variables and coefficients. Here are some key points related to subtraction:

• Subtracting means removing or taking away a specific term from an expression.
• Always ensure that the terms you are going to subtract are aligned correctly.
• Check the operation's signs carefully; subtraction can sometimes confuse the sign.

By understanding the setup and approach to subtraction, the following steps involving like terms become easier.

Combining Like Terms

Once we have rewritten our expression explicitly as \(7x - 3 - 4x\), the next key step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our example, \(7x\) and \(-4x\) are like terms because they both contain the variable \(x\). Here is how you can combine like terms:

• Identify the terms that have the exact same variable and exponent; in this case, both terms are involving only \(x\).
• Add or subtract the coefficients of these terms to combine them. For example, \(7x - 4x = 3x\).

Combining like terms simplifies the expression, reducing the number of terms and making it easier to interpret the overall expression. When done carefully, this process transforms the expression into a simpler and more manageable form.

Simplification of Expressions

Once like terms are combined, the expression becomes \(3x - 3\). Simplification entails rewriting the expression in its simplest form. This means you have reduced the expression as much as possible without losing any information. To simplify expressions:

• Combine all like terms, as done previously.
• Look for opportunities to factor or reduce further, if applicable.
• Simplification doesn't always mean eliminating terms; it's about clarity and efficiency in representation.

After combining like terms, if no further reduction is possible (as in this case), the expression \(3x - 3\) is considered "simplified". The process of simplification provides a cleaner, more precise understanding and representation of the original expression.