Problem 57 - Beginning and Intermediate Algebra

Question

Simplify each expression. $$ \frac{2}{3} x-11+11-\frac{2}{3} x $$

Step-by-Step Solution

Answer

0.

Step 1: Combine Like Terms

Identify and combine the like terms in the expression. In this case, the like terms are the ewline fractional constants involving the variable \(x\) and the constant terms. ewline Combine the terms involving \(x\): ewline ewline \frac{2}{3} x - \frac{2}{3} x ewline

Step 2: Simplify Constants

Next, combine the constant terms: ewline ewline -11 + 11. ewline This combination of constants simplifies to 0.

Step 3: Final Simplification

After combining like terms and simplifying constants, the expression becomes: ewline 0.

Key Concepts

Combining Like Terms

In algebra, combining like terms is an essential step for simplifying expressions. Like terms are terms that have the same variable raised to the same power. For example, in the expression \[ \frac{2}{3}x - 11 + 11 - \frac{2}{3} x \], we have two like terms with the variable \(x\): \[ \frac{2}{3} x \] and \[ -\frac{2}{3} x \.\] To combine these, we simply add or subtract their coefficients. Notice that: \[ \frac{2}{3} x - \frac{2}{3} x = 0 \]. This step helps to reduce the expression by eliminating variable terms that cancel each other out.

Simplifying Constants

Constants are numbers on their own, without any variables involved. When simplifying an expression, we combine the constants just like we combine like terms with variables. In the given problem: \[ -11 + 11 \], we see that adding \(-11\) and \(11\) results in 0. This is because -11 and 11 are opposite numbers, and their sum is zero. When simplifying expressions, always combine the constants and simplify them to get a clearer expression.

Expression Simplification

Expression simplification is the process of making an expression easier to understand and work with. After combining like terms and simplifying constants, we look at the expression to see if it can be reduced further. For the problem \[ \frac{2}{3} x - 11 + 11 - \frac{2}{3} x \], combining like terms \[ \frac{2}{3} x - \frac{2}{3} x = 0 \] and constants \[ -11 + 11 = 0 \] led us to: \[ 0. \]. This means the entire expression simplifies to zero, which indicates no further simplification is necessary. The final simplified form of the expression provides a much clearer and more concise result, which is easier to work with in any further calculations.