Problem 17 - Algebra 1: Concepts and Skills

Question

Identify the like terms in the expression. $$ m+8+6 m $$

Step-by-Step Solution

Answer

The like terms in the expression \(m+8+6m\) are \(m\) and \(6m\).

Step 1: Recognize the terms

Start by identifying the different terms in the expression. A term in an algebraic expression is an numerical constant, a variable, or a constant multiplied by a variable or variables. In the given expression \(m+8+6m\), there are three terms - \(m\), \(8\), and \(6m\).

Step 2: Identify like terms

Next, identify the terms that are considered like terms. Like terms are the terms that contain the same variables raised to the same power. Here, the terms \(m\) and \(6m\) are like terms, since they both have the variable \(m\) raised to the power of 1.

Key Concepts

Identifying Like Terms

When working with algebraic expressions, knowing how to spot like terms is essential. Like terms have the same variable(s) and are raised to the same power. For example, in the expression m+8+6m, we can identify the terms as follows: there's a single m, a constant 8, and then 6m, which also contains the variable m.

Like terms in this case are m and 6m because they both have the variable m to the first power (because any variable without a written exponent has an understood exponent of 1). The constant 8 is not a like term with the others because it does not contain a variable component. When identifying like terms, always look for matching variables and exponents; they are key indications that terms can be combined algebraically.

Algebraic Expression Terms

Algebraic expressions are formed from terms that can be a constant, a variable, or a combination of both, sometimes including exponents or coefficients. Each term is separated by a plus (+) or minus (-) sign. In the expression m+8+6m, we have three distinct terms: m, 8, and 6m.

The term m is a variable term, 8 is a constant because it stands alone without a variable, and 6m involves both a coefficient (6) and a variable (m). Understanding each component within a term is crucial for manipulating and simplifying algebraic expressions effectively.

Combining Like Terms

Once like terms have been identified, they can be combined to simplify the expression. Combining like terms involves summing the coefficients of the like terms while keeping the variable part unchanged. Let's look at the expression m+8+6m again. Now, we already established that m and 6m are like terms because they include the same variable m to the same power.

To combine m and 6m, sum their coefficients. Since m is the same as 1m, when you add 1m and 6m together, you get 7m. Therefore, the expression simplifies to 7m+8, with 7m as the combined like term and 8 remaining unchanged as it is not like the other terms.