Problem 23 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 5 m-7 m-2 m $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: \(5m - 7m - 2m\). Answer: The simplified algebraic expression is \(-4m\).

Step 1: Identify the like terms

In the given expression, the like terms are the ones containing the variable 'm'. So, all three terms in the expression are like terms: \(5m\), \(-7m\), and \(-2m\).

Step 2: Combine the like terms

Now, we will add or subtract the coefficients of the like terms to simplify the expression: \(5m - 7m - 2m\)

Step 3: Simplify the expression

Combine the coefficients of the like terms: \((5 - 7 - 2) m\) Now, calculate the value inside the parentheses: \((-2 - 2) m = -4m\) So, the simplified algebraic expression is: $$ -4m $$

Key Concepts

Like Terms

When dealing with algebraic expressions, it's important to understand the concept of like terms. Like terms are terms that have exactly the same variables raised to the same powers. Because of their similarity, they can be combined together in various operations. In our example, the expression \(5m - 7m - 2m\) involves only one variable, \(m\), which means all the terms are like terms. They each have the variable \(m\), so they share a common characteristic. Identifying like terms is crucial, as it allows us to simplify expressions properly.

When identifying like terms, look for:

• Variables: Make sure the variables are identical in form.
• Powers: The powers of these variables must also match.

This simplification helps in solving algebraic expressions efficiently without changing their value.

Simplification

Simplification is the process of making an algebraic expression easier to understand or work with, by reducing it to its simplest form. For the given expression \(5m - 7m - 2m\), simplification involves combining like terms. This process not only shortens the expression but also maintains its mathematical integrity.

Simplification means:

• Combining all like terms to form a single term.
• Doing arithmetic operations like addition or subtraction on the coefficients of the terms.

In our example, we start by focusing on the numeric parts known as coefficients while keeping the variable \(m\) in place. We perform the arithmetic operation \((5 - 7 - 2)\) leading us to \(-4m\) after computing \(-4\). Simplification helps in reducing clutter and makes complex algebraic problems easier to solve.

Coefficients

Coefficients are the numerical factors that multiply a variable in an algebraic expression. In our expression, \(5m\), \(-7m\), and \(-2m\), the coefficients are 5, -7, and -2 respectively. They tell us how many of the variables we have.

Important points about coefficients include:

• They can be positive or negative.
• Recognizing the coefficients helps in combining like terms.

By focusing on coefficients, we can simplify expressions effectively, as it is the coefficients that we add, subtract, or perform any operation on. In our example, combining the coefficients \(5, -7,\) and \(-2\) helps us reduce the expression to \(-4m\). Understanding coefficients and their role in algebraic expressions is key to mastering algebra.