Problem 74 - Prealgebra

Question

Simplify. $$2 x-11+3 x$$

Step-by-Step Solution

Answer

The expression simplifies to \(5x - 11\).

Step 1: Identify Like Terms

In the expression \(2x - 11 + 3x\), recognize that \(2x\) and \(3x\) are like terms. Like terms are terms that have the same variable raised to the same power.

Step 2: Combine Like Terms

Add the coefficients of the like terms \(2x\) and \(3x\) together. \(2x + 3x = (2+3)x = 5x\).

Step 3: Simplify the Expression

Combine the simplified expression of the like terms with the remaining term. The expression becomes \(5x - 11\). There's no further combination possible since \(5x\) and \(-11\) are not like terms.

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making algebraic expressions easier to work with by combining like terms and eliminating unnecessary elements. The goal is to transform a complex expression into its simplest form, without changing its value. When you simplify an expression like \(2x - 11 + 3x\), you are organizing and reducing it to its most basic components. This process helps in solving equations more easily and allows a clearer understanding of the relationship between different terms. Always remember to look for like terms to combine, which will significantly shrink the expression's complexity.

Like Terms

Like terms are a core concept in algebra because they allow you to combine parts of an expression that have the same variables to the same power. For example, in the expression \(2x - 11 + 3x\), the terms \(2x\) and \(3x\) are like terms because they both contain the variable \(x\) raised to the first power.

• Like terms have the exact same variables and exponents.
• Only the coefficients of like terms are different and need to be combined.
• Numbers without variables are also considered like terms.

By identifying and combining like terms, you can simplify expressions more efficiently, making algebraic manipulation easier.

Coefficients

Coefficients are the numerical factors in terms of an algebraic expression. They tell you how many times a variable is counted in the expression. In the term \(2x\), the coefficient is 2, meaning 2 times \(x\). Similarly, in the term \(3x\), the coefficient is 3. When simplifying expressions, like \(2x + 3x\), you only add or subtract the coefficients:

• The expression \(2x + 3x\) becomes \((2+3)x = 5x\).
• Coefficients combine to scale the variable term.

Understanding coefficients is essential for simplifying expressions and accurately solving algebraic problems.

Algebra Basics

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. At its core, algebra involves expressions, equations, and functions. Simplifying expressions is one of the fundamental skills in algebra because it lays the groundwork for solving more complex equations. Here are a few basics to remember:

• Expressions are combinations of numbers, variables, and operators that show a mathematical phrase.
• You simplify expressions by combining like terms and following mathematical operations rules.
• Understanding variables is key, as they represent unknowns that can be manipulated within an equation.

Mastering these basics ensures you have the foundation necessary for progressing to more advanced topics in mathematics.