Problem 38 - Elementary Algebra

Question

Simplify. $$ 13 x+512 x $$

Step-by-Step Solution

Answer

525x

Step 1 - Combine Like Terms

The expression given is \(13x + 512x\). Both terms are like terms because they have the same variable, \(x\). To simplify this expression, we add the coefficients of \(x\) together. So, we have \(13 + 512\), which simplifies to \(525\).

Step 2 - Write the Simplified Expression

After adding the coefficients obtained in Step 1, attach the result to the variable, which remains \(x\). The simplified expression becomes \(525x\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves combining like terms to make an expression easier to work with. Think of it as tidying up a cupboard by grouping similar items together. In mathematical terms, we look for parts of an expression that share the same variable and operations and then add or subtract them to create a neater expression.

For example, in the expression \(13x + 512x\), both terms involve the variable \(x\). By simplifying this expression, we combine these terms into one single term. This process not only makes the expression easier to manage but can also help you solve equations more efficiently when you encounter them in future problems. Remember, like terms must have exactly the same variable parts for them to be combined.

When dealing with more complex expressions, always make sure to identify and group together all like terms, whether they include variables, numbers, or both, before applying any further operations.

Coefficients

Coefficients are the numbers in front of variables in algebraic expressions. They tell you how many times to multiply the variable. For example, in the term \(13x\), 13 is the coefficient.

The role of coefficients is crucial in simplifying expressions, as they are the parts of the terms you actually perform arithmetic on while combining like terms. When you see an expression like \(13x + 512x\), you know both terms have the same variable \(x\). This similarity allows you to focus on the coefficients, 13 and 512, and simply add them together to combine the terms.

The result of adding these coefficients, 13 plus 512, gives you 525. So, you replace \(13x + 512x\) with \(525x\). This process emphasizes how coefficients provide an efficient way to manage and simplify expressions.

Variable Terms

Variable terms are parts of an algebraic expression that contain variables, such as \(x\), \(y\), or \(z\). A variable represents an unknown or changeable value, which makes it a key component of algebra.

In the expression \(13x + 512x\), the terms \(13x\) and \(512x\) both involve the variable \(x\). These terms are referred to as variable terms. What makes them 'like' terms is the shared variable, \(x\). The idea is to group these 'like' variable terms together to simplify the expression.

Understanding the concept of variable terms is crucial not only for simplifying expressions but also for solving equations and understanding functions. By recognizing which terms share the same variable, you can quickly combine them, reducing complexity and making calculations easier.