Problem 55 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(16-5 m-4 m-2+2 m\)

Step-by-Step Solution

Answer

14 - 7m

Step 1: Combine Like Terms

Identify and group the like terms, which are terms involving the variable \(m\) and constant terms. The expression is \(16-5m-4m-2+2m\).

Step 2: Combine Like Terms with Variable

Add all the coefficients of \(m\). \(-5m - 4m + 2m = -7m\).

Step 3: Combine Constant Terms

Add the constants \(16 - 2 = 14\).

Step 4: Write the Final Expression

Combine the simplified terms from the previous steps to get \(14 - 7m\).

Key Concepts

Like Terms

In algebra, understanding like terms is crucial for simplifying expressions effectively. Like terms are terms that have the exact same variable raised to the same power. For example, in the expression from our exercise, the terms

• -5m,
• -4m,
• and 2m are like terms because they all have the variable 'm'.

The constant terms 16 and -2 are also like terms because they are both numbers without variables. When simplifying expressions, always group and combine like terms to make your calculations easier. This step significantly reduces the complexity of the expression and leads us to the next steps where we deal with coefficients and constants.

Coefficients

Coefficients are the numerical values that multiply the variables in an algebraic expression. In our given solution, the coefficients are:

• -5 in the term -5m,
• -4 in -4m,
• and 2 in 2m.

Identifying and combining these coefficients is essential in simplifying expressions. Let's take a closer look at this process:

• We have -5m and -4m, these can be combined by adding the coefficients: -5 + -4 = -9.
• Then, add the +2 coefficient of the term 2m: -9 + 2 = -7. This gives us -7m.

By combining coefficients, you bring together all terms involving the same variables, which is an essential process in algebraic simplification. This method helps you correctly simplify the expressions step-by-step.

Constants

Constants are terms in an algebraic expression that do not include variables. They are simply numbers. In the given expression 16 - 5m - 4m - 2 + 2m, the constants are 16 and -2. When simplifying an algebraic expression, you should also combine the constants together. Here’s how you do it:

• First, look at 16 and -2. Adding these together: 16 - 2 = 14.
• By combining these constants, you simplify the problem further.

The final step in our exercise combines the simplified variable terms and the simplified constants to give you the final answer: 14 - 7m. Each part—like terms, coefficients, and constants—plays a crucial role in making sure you simplify correctly.