Problem 83 - Introductory Algebra for College Students

Question

Simplify each algebraic expression. $$-5 x+x$$

Step-by-Step Solution

Answer

The simplified expression is \(-4 x\)

Step 1: Identify like terms

In this expression, both terms are similar as they contain the variable 'x'. The coefficients are -5 and 1 respectively.

Step 2: Simplify like terms

Simplify by adding the coefficients when the variables are the same. The sum of -5 and 1 is -4. So, the simplified expression will be \(-4 x\).

Key Concepts

Like Terms

In algebra, understanding like terms is essential for simplifying expressions. Like terms are terms that possess the same variable(s) raised to the same power. For instance, in the expression \(-5x + x\), both terms are considered like terms because they share the same variable, \(x\), and it’s raised to the power of one. Being able to identify like terms allows you to combine them by adding or subtracting, which simplifies the expression.
When simplifying algebraic expressions:

• Assess each term to determine if it has the same variable(s).
• Ensure that the power of the variables matches for all terms you intend to combine.

Stephen when simplifying, recognize that selecting like terms helps reduce complexity, leading to a much simpler mathematical expression.

Coefficients

Coefficients are the numerical factors that multiply the variables in an algebraic expression. In the example \(-5x + x\), -5 and 1 are the coefficients. Coefficients are numbers that appear directly in front of the variables and serve to scale the term by multiplying it.

• The coefficient of \(-5x\) is -5.
• The coefficient of \(x\) is essentially 1, even if the 1 is not explicitly written.

When you hear about simplifying expressions through coefficients, this terminology refers to the process of combining these numbers when dealing with like terms. Simply, you add or subtract the coefficients of like terms while retaining the variable part of the expression. For our example, combining \(-5 + 1\) results in \-4\, and thus the expression becomes \(-4x\). Understanding coefficients is crucial as they significantly impact the overall value of the expression.

Variables

Variables in algebra serve as symbols that can assume various values, allowing expressions and equations to remain general until further defined. For instance, 'x' in the algebraic expression \(-5x + x\) is a variable. It represents an unknown quantity or a quantity that can change, which is central to solving algebraic problems.
Key aspects about variables include:

• They often represent numbers we don't yet know.
• They allow expressions to model real-life scenarios where values are not fixed.
• When combined with coefficients, they form terms like \-5x\ and \+x\.

Variables are what tie the expressions together, and understanding their role will help you manipulate and work with algebraic expressions more effectively. When simplifying expressions, it’s important to keep variables consistent to correctly combine like terms and coefficients.