Problem 20 - Prealgebra

Question

Simplify expression. \(2 x+5 x\)

Step-by-Step Solution

Answer

The expression \(2x + 5x\) simplifies to \(7x\).

Step 1: Identify Like Terms

In the expression \(2x + 5x\), the terms \(2x\) and \(5x\) are like terms because they both contain the variable \(x\). Like terms can be combined by adding their coefficients.

Step 2: Add the Coefficients

To simplify \(2x + 5x\), add the coefficients of the like terms. The coefficient of \(2x\) is 2, and the coefficient of \(5x\) is 5. Therefore, add these coefficients together: \[2 + 5 = 7\]

Step 3: Combine the Terms

After adding the coefficients, combine them with the common variable \(x\). Therefore, after adding the coefficients, \(2x + 5x\) simplifies to \(7x\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a foundational concept in mathematics. They consist of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, or division. Expressions can be as simple as a single number or variable, or as complex as a combination of multiple terms. Each "term" is a part of the expression that could be a standalone number, or a combination of a number and a variable.
For example, in the expression \(2x + 5x\), \(2x\) and \(5x\) are two terms. The letters, like 'x', in algebraic expressions stand for unknown values and they are called variables. Numbers like 2 and 5 in this context are known as coefficients, which are attached to the variables. Understanding how to manipulate these elements is crucial in solving algebra problems.

Simplifying Expressions

Simplifying algebraic expressions is about combining like terms to make an expression easier to understand and work with. Like terms are terms that have the same variables raised to the same power, which means you can directly add or subtract their coefficients.

• Identify terms that can be combined. In the expression \(2x + 5x\), both terms are like terms because they share the same variable 'x'.
• Combine these like terms by adding or subtracting their coefficients. Here you add the coefficients 2 and 5 to simplify the expression to \(7x\).

Simplifying expressions helps in solving equations, analyzing equations, and understanding the relationships between quantities.

Coefficients

A coefficient is a numerical factor that multiplies a variable in an algebraic expression. Coefficients reflect how many parts of the variable are present. If you have the term \(2x\), 2 is the coefficient of \(x\).
Understanding coefficients is important when combining like terms. In our example expression \(2x + 5x\), the coefficients are 2 and 5. Simplifying such expressions involves adding or subtracting these coefficients.
Moreover:

• When coefficients add up, you get a single numeric value in front of the variable, simplifying the operation.
• Coefficients can be positive or negative, affecting how terms interact with one another during simplification.

Recognizing and working with coefficients correctly is a vital skill in algebra.