Problem 62 - Elementary Algebra

Question

Use the commutative properties of addition and multiplication to write equivalent expressions for the following problems. $$ (x+y)(x-y) $$

Step-by-Step Solution

Answer

Question: Use the commutative properties of addition and multiplication to find an equivalent expression to \((x+y)(x-y)\). Answer: The equivalent expressions to \((x+y)(x-y)\) using the commutative properties of addition and multiplication are: 1. \((y+x)(x-y)\) 2. \((x-y)(y+x)\) 3. \((x-y)(y+x)\)

Step 1: Understanding the Commutative Properties of Addition and Multiplication

The commutative property of addition states that changing the order of the numbers being added does not change the sum. In other words, if a and b are any numbers, then a + b = b + a. Similarly, the commutative property of multiplication states that changing the order of the numbers being multiplied does not change the product. In other words, if a and b are any numbers, then ab = ba.

Step 2: Apply the Commutative Property of Addition to the Given Expression

The given expression is \((x+y)(x-y)\). We can observe that the first term within the parentheses is an addition operation, \(x+y\). According to the commutative property of addition, we can reverse the order and rewrite the term as \((y+x)(x-y)\), which is equivalent to the original expression.

Step 3: Apply the Commutative Property of Multiplication to the Given Expression

Now that we have applied the commutative property of addition to the given expression, let's apply the commutative property of multiplication. The given expression is now \((y+x)(x-y)\). Reversing the order of the two factors being multiplied, we rewrite the expression as \((x-y)(y+x)\), which is equivalent to the original expression as well as to the previously obtained expression.

Step 4: Additional Equivalent Expressions

Finally, we can combine the steps above to obtain more equivalent expressions for the given problem. One such possibility would be reversing both the order of the addition term and the order of the multiplication factors. Doing so, we would get the expression \((x-y)(y+x)\). In conclusion, we have found three equivalent expressions for the given problem using the commutative properties of addition and multiplication: 1. \((y+x)(x-y)\) 2. \((x-y)(y+x)\) 3. \((x-y)(y+x)\)

Key Concepts

Equivalent Expressions

Equivalent expressions are different algebraic expressions that have the same value. They may look different but, when simplified, result in the same answer. In our exercise, we've used the commutative property to find expressions equivalent to \((x+y)(x-y)\). Here are a few points to remember about equivalent expressions:

• They give the same result for any value substituted into the variable.
• They can be rewritten using properties like commutative, associative, and distributive laws.
• Finding equivalent expressions is crucial in simplifying complex equations.

By using the commutative property, the expressions \((y+x)(x-y)\) and \((x-y)(y+x)\) were shown to be equivalent to our original expression. This is because rearranging terms via commutative properties does not change the end result of the expression.

Properties of Operations

Properties of operations are rules that apply to arithmetic and algebra, helping simplify calculations and manipulations of expressions. Understanding these properties allows us to transform expressions and equations efficiently.There are several key properties:

• Commutative Property: This property applies to addition and multiplication, allowing the order of numbers to change without affecting the result. For addition: \(a + b = b + a\), and for multiplication: \(ab = ba\).
• Associative Property: Applies to addition and multiplication, stating that the way numbers are grouped does not change their sum or product. For addition: \((a + b) + c = a + (b + c)\), and for multiplication: \((ab)c = a(bc)\).
• Distributive Property: Shows how multiplication is distributed over addition. It states that \(a(b + c) = ab + ac\).

In our problem, we specifically utilize the commutative property to find equivalent expressions, emphasizing how shifting the order of terms does not alter their outcome.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations (like addition and multiplication). They are used to express mathematical relationships and to solve various types of problems. Key components of algebraic expressions include:

• Variables: Symbols (often letters) that represent unknown values or quantities.
• Constants: Fixed numerical values within an expression.
• Operations: Include addition, subtraction, multiplication, and division.

By manipulating algebraic expressions using properties, such as the commutative property discussed in our exercise, we can achieve equivalent forms of expressions. This skill is particularly useful when simplifying expressions or solving equations.