Problem 346 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 7(y-13) $$

Step-by-Step Solution

Answer

7y - 91

Step 1: Identify the Distributive Property

The Distributive Property states that for any numbers or expressions a, b, and c, the expression a(b + c) is equal to ab + ac. Similarly, a(b - c) is equal to ab - ac.

Step 2: Apply the Distributive Property

Here, identify the terms to be distributed. In this case, distribute the 7 to both terms inside the parentheses: \( 7(y - 13) = 7 \times y - 7 \times 13 \)

Step 3: Simplify the Expression

Perform the multiplication for each term: \( 7 \times y = 7y \) \( 7 \times 13 = 91 \).So the expression becomes: \( 7y - 91 \)

Key Concepts

Simplifying Algebraic Expressions

When simplifying algebraic expressions, our goal is to make the expression as simple as possible. This often involves removing parentheses and combining like terms. In this exercise, we use the distributive property to remove the parentheses.
For example, we start with the expression 7(y - 13). This means we need to multiply 7 with both y and -13. Applying the distributive property here will give us an updated expression without parentheses: 7y - 91.
This process makes complicated expressions more manageable and is a fundamental skill in algebra.

Basic Algebra

In basic algebra, we often deal with variables, constants, and the four basic arithmetic operations: addition, subtraction, multiplication, and division.
The exercise uses the distributive property, a key concept in basic algebra, to simplify the expression. This property allows us to break down expressions into simpler parts, making them easier to work with.
Understanding how to manipulate variables and constants using these operations is crucial for solving more complex algebraic problems later on.

Properties of Operations

Properties of operations are rules that make it easier to work with numbers and algebraic expressions.
One of the most important properties is the Distributive Property, which we see in the given exercise. It states that a(b + c) = ab + ac. This allows us to distribute a multiplication over addition or subtraction.
Other useful properties include:

• Commutative Property: a + b = b + a or ab = ba
• Associative Property: (a + b) + c = a + (b + c) or (ab)c = a(bc)

These properties help us rearrange and simplify expressions, making problem-solving more straightforward.