Problem 23 - Prealgebra

Question

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property. $$9 x-x-5-1$$

Step-by-Step Solution

Answer

The simplified expression is \(8x - 6\).

Step 1: Identify Similar Terms

In this expression, identify the similar terms. You have variable terms \(9x\) and \(-x\), and constant terms \(-5\) and \(-1\).

Step 2: Combine Like Terms with Variables

Combine the variable terms. \(9x - x = 8x\). This is done by recognizing that \(9x\) and \(-x\) both have the variable \(x\).

Step 3: Combine Constant Terms

Combine the constant terms. \(-5 - 1 = -6\). This involves simple addition of the numbers \(-5\) and \(-1\).

Step 4: Write the Simplified Expression

After combining, write down the simplified expression which is \(8x - 6\).

Key Concepts

Like Terms

Algebraic expressions often contain parts that can be combined to make them simpler. These parts are known as "like terms." Understanding what like terms are is critical when simplifying expressions. Like terms are terms in an algebraic expression that have the exact same variable raised to the same power. For instance, in the expression \(9x - x - 5 - 1\), the terms \(9x\) and \(-x\) are like terms because they both contain the variable \(x\).In contrast, constant numbers without any variables are also like terms with each other. So, \(-5\) and \(-1\) are like terms in this context. Being able to identify like terms is essential as it helps in managing and simplifying the expression by performing basic arithmetic operations. Recognizing and combining like terms efficiently reduces the expression to its simplest form, making math problems much easier to solve.

Commutative Property

The commutative property is a fundamental concept in mathematics that states that the order in which numbers are added or multiplied does not affect the result. In algebra, this property is especially useful when dealing with expressions that need simplification. It allows you to rearrange terms to a more convenient order for combining like terms.Let's apply this property to our expression \(9x - x - 5 - 1\). Using the commutative property, we can rearrange it to \( 9x - x - 5 - 1 \rightarrow (9x - x) + (-5 - 1) \). This rearrangement helps clearly group like terms together, making it easier to combine them in subsequent steps.It's important to note that while the commutative property applies to addition and multiplication, it does not work for subtraction or division. This distinction is key when manipulating algebraic expressions.

Combining Like Terms

Combining like terms is a vital process in algebra that simplifies expressions and solves equations faster. Once like terms are identified and grouped using the commutative property, they are ready to be combined. This simplifies the algebraic expression.Consider the variable terms \(9x\) and \(-x\). We combine by performing the arithmetic operation: \(9x - x = 8x\). Similarly, with the constant terms \(-5\) and \(-1\), the operation is: \(-5 - 1 = -6\). This logic of combining applies because like terms share a common base, whether a variable or constant, allowing arithmetic operations to be directly related.Ultimately, by effectively combining like terms, the expression \(9x - x - 5 - 1\) is simplified to \(8x - 6\). This step-by-step simplification aids in unpacking math problems, making a complex algebraic process manageable and less daunting for students.