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Chapter 1: Problem 346

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

Short Answer

Expert verified
7y - 91

Step by step solution

01

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.
02

Apply the Distributive Property

Here, identify the terms to be distributed. In this case, distribute the 7 to both terms inside the parentheses: \( 7(y - 13) = 7 \times y - 7 \times 13 \)
03

Simplify the Expression

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In the following exercises, translate the phrases into algebraic expressions. (a) five times the sum of \(3 x\) and \(y\) (b) the sum of five times \(3 x\) and \(y\)

Explain why the sum of -8 and 2 is negative, but the sum of 8 and -2 is positive.

In the following exercises, translate to an algebraic expression and simplify if possible. (a) the difference of -5 and -30 (b) subtract -6 from -13

In the following exercises, translate the phrases into algebraic expressions. (a) the sum of \(3 x^{2} y\) and \(7 x y^{2}\) (b) the product of \(6 x y^{2}\) and \(4 z\) (c) Twelve more than \(3 x^{2}\) (d) \(7 x^{2}\) less than \(63 x^{3}\)

In your own words, state the Associative Property of addition.
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