Problem 11 - Algebra for College Students

Question

State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$7(4)+9(4)=(7+9) 4$$

Step-by-Step Solution

Answer

Distributive Property.

Step 1: Identify the Components of the Expression

The expression given is: \(7(4) + 9(4)\). Notice that both terms, \(7(4)\) and \(9(4)\), include the common factor \(4\).

Step 2: Reorganize Expression to Show Common Factor

Rewrite the expression \(7(4) + 9(4)\) as \((7 + 9) \, \cdot\, 4\). Here, \(4\) is factored out from both terms, showing the multiplication of the sum of the numbers \(7\) and \(9\) by \(4\).

Step 3: Identify the Property Used

Understand that pulling out the common factor \(4\) from \(7\) and \(9\) and writing as \((7 + 9) \, \cdot\, 4\) is based on the Distributive Property. The distributive property states that \(a(b+c)=ab+ac\).

Key Concepts

Commutative Property

The commutative property is one of the fundamental principles in mathematics that makes calculations more flexible. This property applies to addition and multiplication, and it states that changing the order of the numbers you are adding or multiplying does not affect the result.

• For addition, this means: \( a + b = b + a \).
• For multiplication: \( a \times b = b \times a \).

This property is particularly useful for mental math, allowing you to rearrange numbers in a way that might make it easier to calculate. When numbers are swapped in an operation, if the outcome stays the same, the commutative property is at work. In our context, while the original exercise does not directly involve the commutative property, understanding it helps to recognize when factors or sums can be rearranged. This is particularly beneficial when simplifying expressions or solving equations, where reordering can simplify the process.

Factoring

Factoring is the process of breaking down an expression into simpler parts or "factors" that, when multiplied together, will produce the original expression. In the given exercise, recognizing that both terms \(7(4)\) and \(9(4)\) share a common factor allows us to apply factoring effectively. We can factor by:

• Identifying the common factor in all terms, which in this case is \(4\).
• Rewriting the expression by "pulling out" the common factor from each term, leading us to \((7 + 9) \cdot 4\).

This step is not just a simplification, but it also aids in understanding how expressions can be reorganized. Factoring is often an important step in solving equations, as it allows for reducing complexity and identifying underlying structures within an expression. By factoring out the common factor, the given expression becomes simpler and clearer, showcasing the impact of the distributive property.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. This often makes complex expressions easier to work with, interpret, and solve. The process usually includes combining like terms, factoring, using arithmetic operations, and clearing out any unnecessary components.In the example from the exercise, simplifying was achieved by applying the distributive property. Here's how you can simplify an expression conceptually:

• Identify like terms or common factors that can be combined or factored out.
• Apply arithmetic operations such as adding, subtracting, multiplying, or factoring.
• Rewrite the expression to its simplest form, as seen in our example where \(7(4) + 9(4)\) becomes \((7 + 9) \cdot 4\).

A simplified expression is generally shorter and easier to interpret, which is particularly useful for further manipulations or finding solutions to equations. Simplification is essential for efficiency in mathematics, allowing you to focus on core components and operations necessary for solving the task at hand.