Problem 84 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$-6 x+x$$

Step-by-Step Solution

Answer

-5x

Step 1: Identifying Like Terms

Look for terms with the same variable raised to the same power. Here, \(-6x\) and \(x\) are like terms.

Step 2: Combining Like Terms

Combine the like terms by adding or subtracting the coefficients. Here, you are subtracting \(6x\) from \(x\), which can be rewritten as: \(-6x + 1x\)

Step 3: Simplifying

Upon adding the coefficients together, it results in \(-5x\)

Key Concepts

Like Terms

In algebra, like terms are fundamental because they allow us to simplify expressions easily. Like terms are terms within a mathematical expression that have identical variable parts. These variable parts include the variable itself and its exponent. For example, the terms \(2x\) and \(-3x\) are like terms because they both contain the variable \(x\) to the first power.

It's important to note that only the coefficient, the numerical part of the term, has to be different for terms to be considered like terms. The variable and its exponent must remain constant. This means that \(4x^2\) and \(-5x^2\) are like terms, but \(4x\) and \(4x^2\) are not. Getting familiar with identifying like terms is crucial for further simplifying algebraic expressions.

Combining Like Terms

Once like terms are identified, the process of combining them becomes straightforward. Combining like terms is essentially the process of adding or subtracting the coefficients of these terms. Since the variable part is the same, it remains unchanged, and the focus is on the coefficients at this stage.

For example, if you have the terms \(-6x\) and \(x\), first ensure the variable part matches, which it does here as both are \(x\). Next, adjust the coefficients: \(-6\) and \(1\). Combine these by addition or subtraction: \(-6 + 1 = -5\). As a result, combining like terms transforms \(-6x + x\) into \(-5x\).

It's like gathering similar fruits in a basket. You count together only apples with apples, leaving oranges to their own basket.

Coefficients

Coefficients play a significant role in simplifying expressions because they multiply the variable portion of a term. In an algebraic term like \(3x\), the number 3 is the coefficient, and it serves to scale the value of \(x\).

When working with like terms, only coefficients are adjusted, while variables remain fixed. In \(-6x + x\), the coefficients are \(-6\) and \(1\), respectively. To simplify, focus on these numbers: by adding, \(-6 + 1\), you get \(-5\), which means the term becomes \(-5x\).

Understanding coefficients is crucial for analyzing and interpreting algebraic expressions. They affect the steepness of lines in graphing and contribute greatly to solving algebraic equations by helping us find numerical solutions.