Problem 14 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ x+3 x $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \(x + 3x\). Answer: The simplified algebraic expression is \(4x\).

Step 1: Identify the coefficients of the terms

In the given expression, we have 2 terms: 1. \(x\): Here, the coefficient is 1. 2. \(3x\): Here, the coefficient is 3.

Step 2: Add the coefficients of the like terms

Since both terms are like terms (they both contain the variable x), we can add them by adding their coefficients. 1 (coefficient of \(x\)) + 3 (coefficient of \(3x\)) = 4

Step 3: Write the simplified expression

Combining the like terms, we get the simplified expression by multiplying the sum of the coefficients with the variable x. 4 (sum of coefficients) * \(x\) = \(4x\) The simplified algebraic expression is \(4x\).

Key Concepts

Simplifying Expressions

Simplifying expressions means reducing an algebraic expression to its simplest form by combining like terms. This process makes the expression easier to work with and understand. When simplifying, the goal is to condense the expression as much as possible without changing its value. For example, consider the expression \(x + 3x\). Here, we can combine the like terms to form a simpler expression.**Steps to Simplify:**

• Look for terms in the expression that have the same variable or variables raised to the same power. These are called "like terms," and they can be combined.
• Add or subtract the coefficients of these like terms, keeping the variable part the same.
• If necessary, rearrange the terms to highlight those that can be combined for ease of simplification.

In our example, combining the terms results in \(4x\), the simplified form of \(x + 3x\). This step ensures that mathematical problems are easier to solve or further manipulate.

Coefficients

Coefficients are numeric factors that multiply a variable in an algebraic expression. They are crucial for understanding how much of each variable we have. In the expression \(x + 3x\), the coefficient of \(x\) is 1, and the coefficient of \(3x\) is 3.**Key Points About Coefficients:**

• Coefficients can be positive or negative numbers, fractions, or even decimal numbers.
• They tell us "how many" of a variable are present in each term.
• If a variable stands alone, as in \(x\), it typically has an unspoken coefficient of 1, often not written out explicitly.

Understanding coefficients helps in combining like terms because we add or subtract these coefficients while keeping the variables logically organized. After finding the sum of the coefficients, as in our example, we multiply this sum by the variable to get the simplified expression, \(4x\).

Like Terms

Like terms are terms in an expression that contain the same variables raised to the same powers. Only the coefficients of these terms are different. This similarity allows them to be combined. In the example \(x + 3x\), both terms are like terms because they share the same variable \(x\).**Identifying and Combining Like Terms:**

• Like terms must have identical variable parts. For instance, \(2x\) and \(5x\) are like terms, while \(x\) and \(x^2\) are not.
• Simply add or subtract their coefficients to combine them. In our example, adding 1 and 3 results in a coefficient of 4, giving us \(4x\).
• Combining like terms is a key step in simplification and ensures that expressions are written in their most reduced form.

By organizing terms this way, you can make even the most complex algebraic expressions much simpler to understand and solve.