Problem 18 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$|-7| m+|6| m+|-3| m$$

Step-by-Step Solution

Answer

The expression simplifies to \(16m\).

Step 1: Find Absolute Values

Start by finding the absolute values of the coefficients in the expression. The given expression is \(|-7| m + |6| m + |-3| m\). This simplifies to \(7m + 6m + 3m\) since the absolute values of \(-7\), \(6\), and \(-3\) are \(7\), \(6\), and \(3\), respectively.

Step 2: Combine Like Terms

Now, add the coefficients of the like terms, which all have the same variable \(m\). Combine them as follows: \(7m + 6m + 3m = (7 + 6 + 3)m = 16m\).

Key Concepts

Understanding Absolute Value

The concept of absolute value is essential in mathematics, especially when dealing with variables and expressions. Absolute value refers to the distance a number is from zero on the number line, without considering the direction.
In simpler terms, it is always a non-negative number. For instance:

• The absolute value of \(-7\) is \(|-7| = 7\) because it is 7 units away from zero.
• Similarly, \(|6| = 6\) and \(|-3| = 3\).

Absolute values are used to simplify expressions by removing the negative signs and focusing solely on the magnitude of numbers. This is why in the original exercise, the first step was to find the absolute values of coefficients. Doing so converts all values into positive numbers, making it straightforward to combine terms and simplify the expression further.

Basics of Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operations (addition, subtraction, multiplication, etc.). In these expressions, variables are symbols like \(x\) or \(m\) that represent unknown values or quantities.
The parts of an algebraic expression are:

• Coefficients: Numbers that multiply the variables. In \(7m,\) the number 7 is the coefficient.
• Terms: Each part of the expression separated by a plus \(+\) or minus \(-\) sign. For instance, \(7m, 6m,\) and \(-3m\) are all terms.

An understanding of algebraic expressions is essential for simplifying them, as it involves recognizing and operating on these components according to algebraic rules.

Simplifying Expressions by Combining Like Terms

Combining like terms is a fundamental skill in algebra used to simplify expressions. Like terms are terms in an expression that have the same variable raised to the same power. In the context of the original exercise, all terms involve the variable \(m\).
To simplify:

• First, identify the like terms, such as \(7m, 6m,\) and \(3m\).
• Next, add their coefficients: \(7 + 6 + 3 = 16\).
• Finally, attach the common variable to the sum of the coefficients: \(16m\).

This process reduces the complexity of expressions, making them easier to solve or evaluate. It's a key step in many algebra problems, allowing equations to be simplified and solutions to be more easily derived.