Problem 55 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 6(3 x-6)-2(x+1)-17 x $$

Step-by-Step Solution

Answer

The simplified expression is \( -x - 38 \).

Step 1: Distribute within the first parentheses

First, distribute the 6 to each term inside the parentheses: \[ 6(3x - 6) = 6 \cdot 3x - 6 \cdot 6 = 18x - 36 \]

Step 2: Distribute within the second parentheses

Now, distribute the -2 to each term inside the parentheses:\[ -2(x + 1) = -2 \cdot x - 2 \cdot 1 = -2x - 2 \]

Step 3: Combine like terms

Combine the terms from the expressions obtained:\[ 18x - 36 - 2x - 2 - 17x \]Group the x terms and the constant terms:\[ (18x - 2x - 17x) + (-36 - 2) \]

Step 4: Simplify the expression

Simplify each group of terms:\[ (18x - 2x - 17x) = -x \]\[ (-36 - 2) = -38 \]Therefore, the simplified expression is:\[ -x - 38 \]

Key Concepts

Distributive Property

The Distributive Property is a fundamental principle in algebra. It helps in removing parentheses from an algebraic expression by multiplying each term inside the parenthesis by a factor outside. This principle can be expressed as: a(b + c) = ab + ac. In our exercise, we have to distribute the numbers outside the parentheses by each term within the parenthesis. This involves:

• For the first parentheses: multiplying 6 by each term inside, i.e., 6 multiplied by 3x and 6 multiplied by -6, which results in 18x - 36.
• For the second parentheses: multiplying -2 by each term inside, which results in -2x - 2.

Breaking down the expression component by component ensures an accurate application of the distributive property, thus making the expression simpler to handle in subsequent steps.

Combining Like Terms

Combining like terms is crucial in simplifying algebraic expressions. Like terms are those that contain the same variable raised to the same power. For instance, in our exercise:

• The terms 18x, -2x, and -17x are like terms because they all involve the variable x.
• The constants -36 and -2 are like terms because they are both standalone numbers without variables.

By grouping and summing these like terms, you're able to streamline the expression and reduce it to its simplest form. This process aids not only in simplifying the expression but also in keeping algebraic equations manageable, especially when dealing with complex expressions.

Simplifying Expressions

Simplifying expressions is the final goal and involves reducing them to their simplest structure. This means:

• First, you combine the like terms previously organized, which in our case involves: 18x - 2x - 17x resulting in -x.
• Then, simplify the constant terms: -36 - 2 which gives -38.

This results in the simplified algebraic expression: \[-x - 38\]By simplifying, you not only make the expression easier to read and work with, but also prepare it for further calculations or to be used in equations. It's a vital skill in algebra that strengthens problem-solving capabilities and analytical thinking.