Problem 39 - Elementary Algebra

Question

Simplify. $$ -14 x+45+38 x $$

Step-by-Step Solution

Answer

The simplified expression is \(24x + 45\).

Step 1: Group Like Terms

Identify and group the like terms in the expression. In this case, the like terms involve the variable \(x\). The terms are \(-14x\) and \(38x\).

Step 2: Combine Coefficients of Like Terms

Add the coefficients of the like terms together. For the terms \(-14x\) and \(38x\), the coefficients are \(-14\) and \(38\). Calculate \(-14 + 38 = 24\). Thus, the combined term is \(24x\).

Step 3: Write the Simplified Expression

Substitute the combined term back into the expression. The constant term \(45\) remains unchanged. The simplified expression is \(24x + 45\).

Key Concepts

Like terms

In algebra, like terms are terms that have exactly the same variable part. For example, in the expression \(-14x + 38x + 45\), the terms \(-14x\) and \(38x\) are considered like terms because they both contain the variable \(x\). Like terms make it easier to simplify an expression because you only need to combine their coefficients, which are the numbers in front of the variables, while keeping the variable part unchanged. Understanding like terms helps in combining parts of an algebraic expression that are similar, allowing us to simplify expressions more effectively.

Coefficients

Coefficients are the numerical part of the terms in an algebraic expression. They are the numbers that multiply the variables. In the expression \(-14x + 38x + 45\), the coefficients are \(-14\) and \(38\) for the terms that include the variable \(x\). The coefficient of \(x\) indicates how many times \(x\) is being repeated in an equation, whether positive or negative.

• The term \(-14x\) has a coefficient of \(-14\).
• The term \(38x\) has a coefficient of \(38\).
• Constant terms like \(45\) are technically coefficients of \(x^0\).

Grasping the role of coefficients allows you to perform operations like addition or subtraction of like terms to simplify algebraic expressions.

Simplification

Simplification in algebra refers to the process of reducing an expression into its simplest form. This involves combining like terms, eliminating unnecessary terms, and making the expression as concise as possible. For the expression \(-14x + 38x + 45\), simplification steps include:

• Grouping like terms: Identify terms that can be combined (e.g., \(-14x\) and \(38x\)).
• Performing arithmetic operations: Calculate the sum or difference by adding the coefficients of the like terms together. Here, \(-14 + 38 = 24\).
• Rewriting the expression: Substitute the combined term back into the expression alongside any constant terms, resulting in \(24x + 45\).

Simplification makes expressions easier to work with and understand, helping solve algebra problems more efficiently.

Expression

An expression in algebra is a combination of numbers, variables, and operational symbols like + and -. It does not include an equals sign, which differentiates it from an equation. In the given example \(-14x + 38x + 45\), the expression consists of two parts:

• Variable terms: \(-14x\) and \(38x\), which include the variable \(x\).
• Constant term: The number \(45\), which does not vary.

Expressions are used to represent real-world scenarios mathematically, allowing for calculations and predictions. They form the basis for creating equations and functions, which further aids in solving algebraic problems. Understanding expressions is key to mastering more complex algebraic concepts and operations.