Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Identify the rule(s) of algebra illustrated by the statement.\((x+2)-(x+2)=0\)

Step-by-Step Solution

Answer

The rule of algebra being illustrated by \((x+2)-(x+2)=0\) is: 'Subtracting equal quantities gives zero'.

Step 1: Identifying Operation

The statement is \((x+2)-(x+2)=0\). The operation used here is subtraction.

Step 2: Recognizing the Elements

The elements being subtracted - \(x+2\) and \(x+2\) - are identical.

Step 3: Applying the Rule

The rule of algebra under operation here is the rule of subtracting equal quantities. Any number, letter, or algebraic expression subtracted by itself always gives zero as a result.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operation symbols. These expressions represent values that can change based on the values given to the variables. In our example, \(x+2\), the variable is \(x\), and the number 2 is a constant. Together, they form an expression. Algebraic expressions are essential because they allow us to generalize mathematical problems and solve for unknowns.Understanding how each part of an algebraic expression works is crucial.

• Variables: These are symbols, like \(x\), that represent numbers that can change.
• Constants: These are fixed values that do not change, like the 2 in \(x+2\).
• Operators: Symbols such as \(+\) and \(-\) indicate the operations to be performed.

Recognizing these parts helps us to manipulate and simplify expressions effectively, which is fundamental in solving algebraic equations.

Operations in Algebra

Operations in algebra are similar to basic arithmetic; however, they often involve variables. Common algebraic operations include addition, subtraction, multiplication, and division. These operations are crucial when simplifying expressions or solving equations.

• Addition: Involves combining quantities, as seen in the expression \(x+2\).
• Subtraction: This operation takes away one quantity from another, which is key in our example \((x+2)-(x+2)=0\).
• Multiplication: Expanding expressions, such as distributing \(2(x+1)\), involves multiplication.
• Division: Divides one algebraic expression by another.

When performing these operations, it's vital to maintain the balance of equations, ensuring that both sides remain equal. Operations in algebra help us systematically approach and solve problems efficiently.

Subtraction in Algebra

Subtraction in algebra operates much like subtraction in basic arithmetic. However, it involves subtracting not only numbers but also variables. Our example demonstrates this: \((x+2)-(x+2)=0\). Subtraction is particularly important because of the following:

• Identity Element of Subtraction: Any expression subtracted from itself always results in zero, as shown in the example where \((x+2)-(x+2)=0\).
• Inverse Operations: Subtraction can be seen as the inverse of addition. Understanding this helps with rearranging equations.
• Expression Simplification: Subtraction helps in combining like terms which can simplify expressions significantly.

Knowing how to apply subtraction in algebra aids in evaluation and simplification, which are essential steps in solving algebraic problems.