Problem 29 - Elementary and Intermediate Algebra

Question

To prepare for Section 9.3, review simplifying expressions \((\text { Section } 1.8)\) Simplify. [ 1.8] $$ -6(x-2 y)+(6 x-5 y) $$

Step-by-Step Solution

Answer

The simplified expression is \( 7y \).

Step 1: Distribute the \( -6 \)

Start by distributing the \( -6 \) to both terms inside the parenthesis \( (x - 2y) \). This gives us:\[-6(x) -6(-2y) = -6x + 12y\].

Step 2: Write the Expression After Distribution

After distributing the \( -6 \), write the expression as:\[-6x + 12y + 6x - 5y\].

Step 3: Combine Like Terms

Now, combine the like terms. Combine \( -6x \) and \( 6x \) to get: \(-6x + 6x = 0x\). Then, combine \( 12y \) and \( -5y \) to get: \( 12y - 5y = 7y\). This simplifies the expression to:\[0x + 7y = 7y\].

Key Concepts

Distributive Property

To simplify the expression, we first use the distributive property. This property allows us to multiply a single term by each term inside a parenthesis. For example, in the expression [-6(x-2y)], we multiply -6 by both x and -2y. Distributing -6, we get: -6x - 6(-2y). Simplifying further, -6(-2y) = 12y. After distribution, the first part of our expression becomes: -6x + 12y.

Combining Like Terms

After using the distributive property, the expression looks like this: -6x + 12y + 6x - 5y. Now, we need to combine like terms to simplify the expression further. Like terms are terms that have the same variables raised to the same power. Here, -6x and 6x are like terms, as are 12y and -5y. To combine them, we add or subtract their coefficients: -6x + 6x = 0x and 12y - 5y = 7y. Now our expression is: 0x + 7y.

Algebraic Simplification

The final step is to simplify the expression completely. From our previous step, we have 0x + 7y. Since 0x is just zero, we are left with 7y. This is our simplified expression. To summarize, using the distributive property, combining like terms, and simplifying the result helps us transform complex algebraic expressions into their simplest forms.