Problem 69 - Intermediate Algebra

Question

Simplify by combining like terms. See Example 5 . $$-8 x+5 x-(-x)$$

Step-by-Step Solution

Answer

The simplified expression is \(-2x\).

Step 1: Identify Like Terms

Start by identifying like terms in the expression. Like terms are those that contain the same variables raised to the same power. In \(-8x + 5x - (-x)\), all terms involve the variable \(x\).

Step 2: Rearrange and Simplify Signs

Rearrange the terms for clarity: \(-8x + 5x + x\). Notice that the \(-(-x)\) becomes \(+x\) because subtracting a negative is the same as adding.

Step 3: Combine Like Terms

Combine the like terms by adding or subtracting their coefficients: \(-8x + 5x + x = (-8 + 5 + 1)x\).

Step 4: Simplify Coefficients

Calculate the sum of the coefficients: \(-8 + 5 + 1 = -2\).

Step 5: Write the Final Expression

Substitute the sum of the coefficients back into the expression, yielding \(-2x\).

Key Concepts

Combining Like Terms

When working with algebraic expressions, one fundamental skill is combining like terms. This process helps simplify expressions and make them easier to work with, especially when solving equations. Like terms are terms that contain the same variable raised to the same power. Here are a few crucial points about combining like terms:

• Identify the like terms: Look for terms that contain the same variables. For instance, in the expression \(-8x + 5x - (-x)\), all the terms have the variable \(x\).
• Simplify signs: Sometimes, expressions include subtraction of negatives, which must be refactored as additions. This is evident in our example, where \(-(-x)\) becomes \(+x\).
• Combine the coefficients: Once like terms are gathered, add or subtract their coefficients. Here, the coefficients are \(-8\), \(5\), and \(1\), resulting in \(-2\) when combined.

By transforming complex terms into a simpler form, you are more prepared for further algebraic manipulations.

Algebraic Expressions

Algebraic expressions play a central role in algebra. They are combinations of numbers, variables, and operators such as addition or subtraction. Understanding these expressions can reveal insights into how to solve more complex problems. Here's a closer look at what defines algebraic expressions:

• Terms: These are the building blocks of expressions, which can be constants (just numbers), variables (like \(x\)), or the product of numbers and variables.
• Operations: Operations indicate what you are doing with the terms, such as adding, subtracting, multiplying, or dividing them.
• Structure: Expressions can be simple, like \(3x\), or more complex, such as \(-8x + 5x - (-x)\). Recognizing the structure helps determine the next steps in problem-solving.

Whether simplifying, factoring, or solving, understanding algebraic expressions is essential for progress in algebra and beyond.

Linear Equations

Simplifying expressions often sets the stage for solving linear equations. Linear equations form the backbone of many algebraic processes and can describe a wide variety of scenarios. Linear equations have a few distinct characteristics worth noting:

• Each term is either a constant (a number on its own) or the product of a constant and the first power of a variable (like \(x\)).
• The equation represents a straight line when graphed.
• The goal is typically to find the value of the variable that makes the equation true.

For example, after simplifying the expression \(-8x + 5x - (-x)\) to \(-2x\), one might solve an equation like \(-2x = 10\) by isolating \(x\). Such skills are critical as you advance in mathematics, as it bridges simple arithmetic to more complex operations and applications.