Problem 29 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ 3(2 x-4 y)-2(x+9 y) $$

Step-by-Step Solution

Answer

The simplified expression is \(4x - 30y\).

Step 1: Apply the Distributive Property

Begin by applying the distributive property to both parts of the expression: - Distribute 3 into the terms inside the first set of parentheses: \[ 3 \times (2x - 4y) = 6x - 12y \] - Distribute -2 into the terms inside the second set of parentheses: \[ -2 \times (x + 9y) = -2x - 18y \] So, the expression becomes: \[ 6x - 12y - 2x - 18y \]

Step 2: Combine Like Terms

Next, combine like terms in the expression:- Combine the "x" terms: \[ 6x - 2x = 4x \]- Combine the "y" terms: \[ -12y - 18y = -30y \] Thus, the simplified expression is:\[ 4x - 30y \]

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that helps in breaking down expressions inside parentheses. Whenever you see a number or a variable outside a set of parentheses, you can distribute this term to each individual term inside the parentheses. This means you multiply the outside term with each term inside separately. For example, in the expression \(3(2x - 4y)\), you would multiply 3 by each term inside the parentheses:

• 3 multiplied by \(2x\) gives \(6x\)
• 3 multiplied by \(-4y\) gives \(-12y\)

This converts the expression to \(6x - 12y\). Applying the distributive property to \(-2(x + 9y)\) would result in \(-2x - 18y\). Applying this property simplifies expressions and sets up the equation for further steps in simplification.

Simplifying Expressions

Simplifying expressions refers to the process of rewriting them in a more concise form without altering their value. After distributing any terms outside parentheses, you often find yourself with a new expression that combines similar or identical variable terms. This transformation makes expressions easier to understand and work with in calculations.
It is important to ensure that all distributed terms are accounted for and all operations are correctly signed according to the signs that preceded them. For example, after applying the distributive property to the problem \(3(2x - 4y) - 2(x + 9y)\), the expression simplifies to \(6x - 12y - 2x - 18y\). This form is now set up for the next step of combining like terms.

Combining Like Terms

Combining like terms involves merging terms that have identical variable parts. This step is essential to further condense and simplify the expression. When combining like terms, focus on aligning terms that share the same variables and their respective powers. For instance, in the expression \(6x - 12y - 2x - 18y\):

• The terms involving \(x\) are 6x and -2x.
• The terms involving \(y\) are -12y and -18y.

Now, by combining them:

• Combine \(6x - 2x\) to get \(4x\).
• Combine \(-12y - 18y\) to get \(-30y\).

This results in \(4x - 30y\). Combining like terms is a crucial step in reducing the expression to its simplest form, making it more comprehensible and manageable.