Problem 64 - Algebra A Combined Function

Question

Perform each indicated operation. Don't forget to simplify if possible. Subtract \(m-3\) from \(2 m-6\)

Step-by-Step Solution

Answer

The result of the subtraction is \(m - 3\).

Step 1: Write down the expressions

We need to subtract the expression \(m - 3\) from \(2m - 6\). This can be written as \((2m - 6) - (m - 3)\).

Step 2: Distribute the negative sign

Apply the negative sign to each term in the second expression: \((2m - 6) - m + 3\).

Step 3: Combine like terms

Combine the \(m\) terms: \(2m - m = m\) and combine the constant terms: \(-6 + 3 = -3\). The expression simplifies to: \(m - 3\).

Step 4: Simplify the expression

The simplified expression from the subtraction is \(m - 3\). Since this cannot be simplified further, it is the final answer.

Key Concepts

Subtraction of Expressions

When we talk about subtracting expressions, we're focusing on pinpointing terms we need to deduct from another set of terms. The key here is understanding that subtraction influences each and every term in the expression we're subtracting. For the expression given in the exercise, we start by identifying what needs to be subtracted. We're asked to subtract \(m - 3\) from \(2m - 6\). Simply put, this looks like finding the difference:

• Write down: \((2m - 6) - (m - 3)\).
• Place the entire expression \((m - 3)\) in parentheses to remember the subtraction applies to both terms.

Next, distribute the negative sign across the terms in the expression we are subtracting. This step changes the sign of each term in the parentheses, turning our expression into \((2m - 6) - m + 3\). This sign change is crucial because it sets the stage for the next steps in the problem.

Simplification

Simplifying an expression involves making it as concise as possible while keeping its value unchanged. It's about removing any unnecessary complexity without altering the meaning. After distributing the negative sign, we obtained the expression \((2m - 6) - m + 3\). To simplify, we first need to identify any like terms that we can combine. Simplification includes:

• Eliminating parentheses if possible.
• Combining like or similar terms, which are terms that have the same variable raised to the same power.

By combining

• the variable components: \(2m - m\)
• and the constant terms: \(-6 + 3\)

We achieve a cleaner, more straightforward expression: \(m - 3\). This is simpler to work with and interpret.

Combining Like Terms

Combining like terms is a fundamental part of simplifying expressions in algebra. It involves identifying and summing all terms that have the same variable and exponent. In our exercise, after we distributed the negative sign, we grouped the like terms:

• The like terms \(2m\) and \(-m\) are combined by simply subtracting the coefficients: \(2 - 1 = 1\), resulting in \(m\).
• The constant terms \(-6\) and \(+3\) are likewise added together, which leads to: \(-6 + 3 = -3\).

This conceptual process of combining like terms not only makes the expression more tidy but also easier to manage and work with in subsequent mathematical operations. Essentially, it reduces complexity and helps in maintaining precision in mathematical calculations.