Problem 16 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 2 x+13-1 x $$

Step-by-Step Solution

Answer

The simplified expression is \(x + 13\).

Step 1: Identify Like Terms

Look at the expression \(2x + 13 - 1x\). We see that there are terms involving \(x\) (which are \(2x\) and \(-1x\)), and one constant term (which is \(+13\)). Our first goal is to combine the like terms, which are the terms involving \(x\) together.

Step 2: Combine Like Terms

Combine the terms with \(x\): \(2x - 1x\). This can be done by subtracting the coefficients of \(x\). So, \(2x - 1x = (2 - 1)x = 1x\). Therefore, \(2x - 1x = x\).

Step 3: Write the Simplified Expression

Now, after combining the like terms, we rewrite the expression using the simplified terms. We have combined \(2x - 1x\) into \(x\), and we still have the constant \(+13\). Thus, the simplified expression is \(x + 13\).

Key Concepts

Understanding Like Terms

In algebra, "like terms" refer to terms in an expression that contain the same variables raised to the same power. When simplifying expressions, identifying like terms is crucial because they can be combined to simplify the expression. For example, in the expression \(2x + 13 - 1x\), the terms \(2x\) and \(-1x\) are like terms. They both contain the variable \(x\) to the first power. To combine like terms, you simply need to add or subtract the coefficients—that is, the numbers in front of the variable. So, \(2x - 1x\) simplifies to \((2-1)x = 1x = x\). Recognizing like terms quickly can make simplifying algebraic expressions much more straightforward.

The Constant Term

A constant term is a term in an algebraic expression that does not contain any variables. It's essentially a standalone number in the expression. In our example, the number \(13\) is a constant term because it isn't linked to any variable like \(x\) or any power of a variable. Constant terms remain unchanged while simplifying an expression because there are no other terms to combine them with, as they do not share a common variable component. Therefore, in the expression \(2x + 13 - 1x\), the constant term \(+13\) remains as is after the like terms are combined, resulting in the final simplified expression \(x + 13\).

Role of Coefficients

Coefficients are the numerical part of the terms with variables in an algebraic expression. They are the numbers that multiply the variable. In the expression \(2x + 13 - 1x\), \(2\) and \(-1\) are coefficients. They tell you how many times the variable is being multiplied. When simplifying expressions, you often focus on the coefficients of the like terms to combine them. For instance, to simplify \(2x - 1x\), you subtract the coefficients: \(2 - 1 = 1\), which gives \(1x\) or simply \(x\). Understanding how to work with coefficients is essential to mastering algebraic simplification.