Question

Simplify the expression. Assume all variables are positive. \(\frac{6 x}{2}+4 x\)

Short answer

The simplified expression is 7x.

Step-by-step solution

Perform Division

According to the order of operations, division should be performed first. Divide 6x by 2 to get 3x. The expression becomes 3x + 4x.

Perform Addition

The two terms of the equation are like terms (they both have the variable x), and so can be added together. Add 3x and 4x to get 7x.

Key concepts

Order of Operations

In algebra, it is crucial to follow the correct sequence of operations to simplify an expression accurately. This sequence is known as the order of operations, or by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). When simplifying algebraic expressions, identify the operations present and work through them in the correct order. For example, in the expression \(\frac{6 x}{2}+4x\), division should be performed before addition. First, 6x is divided by 2 to receiving 3x, and only then is 4x added to the result.

Like Terms

In algebra, like terms are terms that have the exact same variable part, raised to the same power. In other words, they're terms that differ only by their numerical coefficient. An example of like terms would be 3x and 4x, as both terms have x as the variable. Simplifying expressions often involves combining like terms. For instance, once you have the expression 3x + 4x, you can add them together to get 7x because of their 'likeness' in terms of the variable factor.

Variable Expression

A variable expression is an algebraic expression that includes variables, numbers, and operations. The variables represent unknown values and are usually denoted by letters such as x, y, or z. In the given example \(\frac{6 x}{2}+4x\), 'x' is the variable within the expression. Simplifying a variable expression involves applying the order of operations and combining like terms to write the expression in its simplest form. The goal is to make the expression easier to work with, especially when it comes to further algebraic manipulations or solving equations.

Algebraic Addition

The process of algebraic addition is fundamental in simplifying algebraic expressions. It entails adding or subtracting terms that are like terms. When performing algebraic addition, it's important to keep in mind that only the coefficients of like terms are added together; the variable part remains unchanged. For example, in the expression 3x + 4x, you are adding the coefficients 3 and 4 while maintaining the variable x, resulting in a sum of 7x. This simplification makes algebraic expressions cleaner and prepares them for further mathematical operations or problem-solving.