Problem 59 - Elementary Algebra

Question

Use the commutative properties of addition and multiplication to write equivalent expressions for the following problems. $$ 2(a-1) $$

Step-by-Step Solution

Answer

Question: Write equivalent expressions for the given expression \(2(a-1)\) using the commutative properties of addition and multiplication. Answer: The equivalent expressions for the given expression are \(-2 + 2a\) and \(-2 + a2\).

Step 1: Identify the factors in the expression

The expression is \(2(a-1)\). The factors are \(2\) and \((a-1)\).

Step 2: Apply the commutative property of multiplication

We can rearrange the factors in the expression using the commutative property of multiplication. This means, \(2(a-1)\) can be written as \((a-1)2\). Now let's apply the distributive property to expand both expressions and then check if we can apply the commutative property of addition.

Step 3: Apply the distributive property

The distributive property allows us to multiply each term inside the parentheses by the term outside the parentheses. For the expression \(2(a-1)\), it becomes: $$ 2(a-1) = 2a - 2 $$ For the expression \((a-1)2\), it becomes: $$ (a-1)2 = a2 - 2 $$

Step 4: Apply the commutative property of addition

Now, we will apply the commutative property of addition to rearrange the terms in both expressions: For \(2a - 2\), we can rearrange the terms: $$ 2a - 2 = -2 + 2a $$ For \(a2 - 2\), we can also rearrange the terms: $$ a2 - 2 = -2 + a2 $$ Now we have found equivalent expressions for the given expression using the commutative properties of addition and multiplication: - \(2(a-1) = 2a - 2 = -2 + 2a\) - \((a-1)2 = a2 - 2 = -2 + a2\)

Key Concepts

Equivalent Expressions

Equivalent expressions may look different but they represent the same value. This is an essential concept in algebra that allows us to rewrite mathematical expressions for easier manipulation. For example, the expression \(2(a-1)\) can be transformed using various algebraic properties, but the essential value remains unchanged.

When working with equivalent expressions, you often use properties like the commutative and distributive properties. We'll explore this in detail in the next sections. Understanding equivalent expressions helps in simplifying complex equations, solving for variables, and checking the consistency of mathematical expressions.

The goal is to recognize that despite appearances, the values of the expressions stay equivalent through careful application of algebraic properties.

Distributive Property

The distributive property is a fundamental concept in algebra. It allows you to multiply a single term by all terms inside a set of parentheses. This property is key to transforming expressions and making them easier to work with.

For instance, in the expression \(2(a-1)\), the distributive property lets us write it as \(2 \times a - 2 \times 1\), which simplifies to \(2a - 2\). This shows how multiplication distributes over subtraction.

• It ensures that each term inside the bracket is multiplied by the factor outside.
• Helps eliminate parentheses and simplifies expressions.

Using the distributive property effectively can make solving algebra problems much more straightforward by breaking down complex expressions into simpler parts.

Multiplication

Multiplication is one of the basic operations in arithmetic and serves as a building block for many algebraic concepts. In expressions like \(2(a-1)\), multiplication connects the number \(2\) with the expression \((a-1)\).

When using multiplication in the context of algebra, properties like the commutative property become particularly useful. This property states that the order in which you multiply numbers does not affect the product. So, \(2(a-1)\) is equivalent to \((a-1)2\) because of this property.

• Order of multiplication doesn't impact the final result due to the commutative property.
• Essential in expanding and simplifying algebraic expressions.

Understanding these underlying properties of multiplication not only helps in computations but also aids in grasping more advanced mathematical concepts.