Problem 41 - Beginning and Intermediate Algebra

Question

Decide whether each statement is an example of a commutative, an associative, an identity, \(a n\) inverse, or the distributive property. $$ 4(2 x)+4(3 y)=4(2 x+3 y) $$

Step-by-Step Solution

Answer

Distributive property.

Step 1: Identify the Operation

Examine the given equation: \[ 4(2x) + 4(3y) = 4(2x + 3y) \] Identify if the operation applied here involves addition, multiplication, or both combined.

Step 2: Identify Properties

List the properties to be checked against. These are: commutative property, associative property, identity, inverse, and distributive property.

Step 3: Look for Distribution

Notice that the left-hand side of the equation is distributing the number 4 to each term inside the parentheses. It changes from \[ 4(2x) + 4(3y) \] to \[ 4(2x + 3y) \].

Step 4: Verify Distributive Property

The operation shown is distributing the 4 across terms within parentheses and combining them into a single parentheses expression, fitting the distributive property definition.

Key Concepts

distributive property

The distributive property is a fundamental algebraic principle. It states that multiplying a number by a sum is the same as doing each multiplication separately. This means you distribute the multiplication over each term within the parentheses.
In the exercise, the equation is: \[ 4(2x) + 4(3y) = 4(2x + 3y) \].
Here, the left-hand side distributes the 4 to both \(2x\) and \(3y\), simplifying to the right-hand side where the 4 multiplies the sum inside the parentheses.
The distributive property formula is: \[ a(b + c) = ab + ac \].In our case: \[ 4(2x + 3y) = 4(2x) + 4(3y) \].
Understanding this property helps break down complex algebraic expressions into simpler parts.

algebraic properties

Algebra relies on several core properties to solve equations effectively. These include:

• Commutative Property: Order doesn't matter (e.g., \(a + b = b + a\)).
• Associative Property: Grouping doesn't matter (e.g., \((a + b) + c = a + (b + c)\)).
• Identity Property: Adding 0 or multiplying by 1 leaves the original number unchanged (e.g., \(a + 0 = a\) or \(a \cdot 1 = a\)).
• Inverse Property: Adding a number's negative or multiplying by its reciprocal returns the identity (e.g., \(a + (-a) = 0\) or \(a \cdot \frac{1}{a} = 1\)).

The given statement in the exercise is an example of the distributive property. Comparing these helps solidify recognition and correct application of each property.

polynomial expressions

Polynomial expressions consist of variables and coefficients combined using addition, subtraction, and multiplication.
When simplifying polynomials using the distributive property, you distribute a number or variable outside the parentheses to each term inside:
For instance, \(3(x + y + z)\) distributes to \(3x + 3y + 3z\).
In our exercise, we see polynomial terms: \(2x\) and \(3y\). Applying the distributive property, \(4(2x + 3y)\) expands to \(4 \cdot 2x + 4 \cdot 3y = 8x + 12y\).
Breaking down polynomial expressions using this property makes solving them straightforward and manageable.