Problem 50 - Prealgebra

Question

Combine like terms. $$6 x+20 x$$

Step-by-Step Solution

Answer

The combined expression is \(26x\).

Step 1: Identify Like Terms

The expression given is \(6x + 20x\). In this expression, both terms \(6x\) and \(20x\) are like terms because they have the same variable \(x\) raised to the same power of 1. Like terms can be combined by adding their coefficients.

Step 2: Add Coefficients

To combine like terms, add the coefficients of \(6x\) and \(20x\). This means calculating \(6 + 20\).

Step 3: Write Combined Terms

After finding the sum of the coefficients in Step 2, which is \(26\), multiply this by the common variable part, \(x\). This gives you the combined expression as \(26x\).

Key Concepts

Understanding Like Terms

In algebra, like terms are terms that have the same variables raised to the same power. This implies that only the coefficients (the numerical part) differ in like terms. For example, in the expression \(6x + 20x\), both terms contain the variable \(x\) raised to the first power. Thus, they are considered like terms.Identifying like terms is crucial because it indicates which terms can be combined to simplify an expression. Like terms can include:

• Terms with identical variables and exponents, such as \(4y\) and \(5y\).
• Constant terms, such as 7 and 3, since they have no variable part.

By spotting like terms, you make it easier to work through algebraic expressions and equations. Remember, however, that terms like \(x\) and \(x^2\) are not like terms, as their exponents differ.

How to Combine Like Terms

Combining like terms simplifies an algebraic expression, making it easier to handle. To combine like terms, you simply add or subtract their coefficients while keeping the variable part unchanged. For instance, in the expression \(6x + 20x\), both terms are like terms because they contain the same variable \(x\). By adding their coefficients, which are 6 and 20:

• Calculate \(6 + 20 = 26\).
• Attach the common variable part, \(x\), to the result.

Thus, the expression simplifies to \(26x\). Combining like terms reduces clutter in equations, leading to more straightforward problem-solving. It’s an essential tool for simplifying algebraic expressions.

Role of Coefficients in Algebraic Expressions

Coefficients are the numerical parts of terms in an algebraic expression. They are the multipliers that accompany the variable parts of the terms. For example, in the term \(6x\), the coefficient is 6.Understanding coefficients is vital because they determine how many times the variable part is counted. Here’s what to remember about coefficients:

• A coefficient provides the scalar quantity for the variable.
• In expressions like \(6x\), modifying the coefficient alters the term's value proportional to the variable.
• In terms without a visible number, such as \(x\), the coefficient is understood to be 1.

In our given example, the coefficients of \(6x\) and \(20x\) are 6 and 20, respectively. By adjusting coefficients, you manipulate term quantities, aiding in the simplification and resolution of algebraic expressions.