Problem 94 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-2(3 r-4)-(6-r)+2 r-5\)

Step-by-Step Solution

Answer

-3r - 3

Step 1: Distribute the -2

Apply the distributive property to the term \(-2(3r - 4)\): \{-2 * 3r + (-2) * (-4) = -6r + 8}.

Step 2: Distribute the negative sign

Apply the distributive property to the term \( -(6 - r) \): \{-6 + r}.

Step 3: Combine like terms

Combine all like terms: \(-6r + 8 - 6 + r + 2r - 5 \). Group the r terms and the constant terms: \{(-6r + r + 2r) + (8 - 6 - 5)}. Simplify the grouped terms: \{-3r - 3}.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions by distributing or multiplying a single term across terms inside a parenthesis. For example, in the expression \(-2(3r - 4)\), we apply the distributive property by multiplying \-2\ with each term inside the parenthesis. This gives us:
\(-2 * 3r = -6r\) and \(-2 * (-4) = 8\). Combining these results, we get \(-6r + 8\).
Distributing makes larger expressions easier to handle by breaking them down. Remember to keep track of positive and negative signs when distributing terms.

Combining Like Terms

Combining like terms is the process of simplifying an expression by merging terms that have the same variable part. In the expression \(-6r + 8 - 6 + r + 2r - 5\), like terms share the same variable: \r\ terms and constant terms. Let's identify and group them:

• \