Problem 67 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ 3 x+7 x $$

Step-by-Step Solution

Answer

The simplified expression is \(10x\).

Step 1: Identify Like Terms

First, recognize the like terms in the expression. Like terms are terms that have the same variable raised to the same power. Here, the terms are \(3x\) and \(7x\), which are both like terms because they both have the variable \(x\).

Step 2: Add the Coefficients

To combine the like terms, add their coefficients. The coefficient of \(3x\) is \(3\), and the coefficient of \(7x\) is \(7\). Add these coefficients together: \(3 + 7 = 10\).

Step 3: Rewrite the Expression

Use the sum of the coefficients with the variable \(x\) to rewrite the expression. This gives us \(10x\).

Key Concepts

Coefficients

In algebra, coefficients are the numerical parts of terms that contain variables. They tell you how many times to take the corresponding variable. For example, in the term \(3x\), the number \(3\) is the coefficient. Similarly, in \(7x\), the coefficient is \(7\). Coefficients are crucial because they determine the fraction of the value represented by the variable. This is particularly important when simplifying expressions by combining like terms.

When combining like terms, only the coefficients are added or subtracted together, while the variable part remains unchanged. In the expression \(3x + 7x\), you add the coefficients \(3\) and \(7\) to get \(10x\). The variable part \(x\) stays the same, and now you've got a cleaner, more manageable expression.

Remember, when no number appears in front of a variable, as in \(x\), the coefficient is assumed to be \(1\). This understanding of coefficients helps in manipulating and simplifying algebraic expressions.

Like Terms

Like terms are terms in an algebraic expression that have the same variable(s) raised to the same power. Recognizing like terms is the first step to simplifying many algebraic expressions.

For example, in the expression \(3x + 7x\), both terms are like terms because they both include the variable \(x\) to the same power (which is 1). Only the coefficients are different. By combining like terms, you make expressions simpler, reducing potential errors when evaluating or solving them.

A handy tip is to also watch out for terms with the same variable order. For example, \(2xy \) and \( -5xy\) are also like terms because both have the variables \(x\) and \(y\) multiplied together the same way, with no additional powers or inequalities placed between them.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (like addition or multiplication). They can range from simple combinations like \(3x + 7\) to more complex ones involving multiple variables and operations.

Understanding the structure of algebraic expressions is fundamental when trying to simplify them, as seen in the exercise with \(3x + 7x\). You identify the components like coefficients and like terms, which are integral parts of the expression.

Algebraic expressions don't have an equality sign like equations, meaning they cannot be directly "solved." Instead, they can be simplified. Simplifying makes these expressions easier to work with. The process involves combining like terms, as well as often factoring or distributing elements within the expression. This simplification is beneficial in solving equations or inequalities later when you're working under constraints or specific conditions.