Problem 41 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ -2(y-z) $$

Step-by-Step Solution

Answer

The expression \(-2(y-z)\) simplifies to \(-2y + 2z\).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers or expressions, \( a(b+c) = ab + ac \). In this case, we have an expression \( -2(y-z) \), which can be rewritten as \( -2 \times (y - z) \). To distribute \(-2\), multiply \(-2\) with each term inside the parentheses.

Step 2: Multiply Each Term

Begin multiplying \(-2\) with each term inside the parentheses one by one. 1. First, multiply \(-2\) by \(y\): \(-2 \times y = -2y\)2. Next, multiply \(-2\) by \(-z\): \(-2 \times (-z) = 2z\).

Step 3: Write the Expression without Parentheses

Combine the results from the multiplication. The expression \(-2(y-z)\) becomes \(-2y + 2z\) after applying the distributive property.

Step 4: Simplify the Expression

The expression \(-2y + 2z\) is simplified and does not have like terms that can be further combined. Therefore, \(-2y + 2z\) is the final simplified form.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are fundamental in algebra and help represent relationships among numbers in a concise way. For instance, \[-2(y - z)\]is an algebraic expression. It consists of numbers \(-2\), variables \(y\) and \(z\), and operations within the parentheses. Understanding algebraic expressions involves knowing how to manipulate them using rules and properties like the distributive property.

Important aspects to consider with algebraic expressions include:

• Variables: These are symbols, like \(y\) and \(z\), that stand for unknown values.
• Operations: Indicated by signs such as plus \(+\), minus \(-\), times \(\times\), or divide \(\div\).
• Constants: Numbers such as \(-2\) that do not change.

When you learn to understand and work with algebraic expressions, you're preparing to tackle more complex algebraic problems with confidence.

Simplifying Expressions

The process of simplifying expressions involves reducing an expression to its simplest form. This often means removing parentheses and combining like terms when possible. In the expression \(-2(y - z)\),the goal is to distribute \(-2\) across the terms within the parentheses.

Steps to simplify:

• Apply the distributive property: Distribute by multiplying each term inside the parentheses by \(-2\). This gives us \(-2 \times y = -2y\)
• Then, \(-2\) multiplied by \(-z\) results in \(2z\). This is because multiplying two negative numbers results in a positive.
• Combine the results: The expression \(-2y + 2z\) does not have any like terms, so it is already in its simplest form.

Simplifying ensures the expression is easier to work with and understand, which is essential for solving algebraic equations.

Negative Numbers

Negative numbers are numbers less than zero. They are located on the left side of the number line and have a distinct role in algebra, especially when operations like multiplication or addition are involved.

When working with negative numbers, consider the following:

• Multiplying two negative numbers yields a positive result. For example, \((-2) \times (-z) = 2z\).
• Adding or subtracting negative numbers can alter the sign of the result. In \(-2(y-z)\), multiplying \(-2\) across \(y\) and \(-z\), changes the sign of \(z\) from negative to positive.

Negative numbers are integral to understanding how to distribute and simplify expressions, as the sign impacts how terms are combined or simplified. Mastering operations with negative numbers unlocks greater understanding in algebra and helps in accurately managing these algebraic expressions.