Problem 8 - Elementary Algebra

Question

Simplify. $$ 10(6 x-9)-(80 x-35) $$

Step-by-Step Solution

Answer

The simplified expression is \(-20x - 55\).

Step 1: Distribute the 10

Apply the distributive property to the expression \(10(6x-9)\). This means multiplying 10 by each term inside the parenthesis: \(10 \times 6x\) and \(10 \times (-9)\). The result is \(60x - 90\).

Step 2: Simplify the Second Group

Since there are no parentheses in the expression \(- (80x - 35)\), distribute the negative sign through the terms: \(-80x + 35\).

Step 3: Combine Like Terms

Now combine the results of each part: \(60x - 90 - 80x + 35\). Combine the like terms \(60x - 80x\) which gives \(-20x\), and then combine the constant terms \(-90 + 35\) to get \(-55\). The simplified expression is \(-20x - 55\).

Key Concepts

Distributive Property

In algebra, the distributive property is a useful tool for simplifying expressions. It involves multiplying a single term by each term within a set of parentheses and helps to "distribute" multiplication across addition or subtraction.
One classic example involves distributing in expressions like this:

• Given an expression like \(10(6x-9)\), apply the distributive property by multiplying the 10 with each term inside the parentheses.
• Calculation: \[10 \times 6x = 60x\]
• Calculation: \[10 \times (-9) = -90\]
• The result simplifies the expression to \(60x - 90\).

By removing parenthetical groupings, the distributive property makes an algebraic expression easier to manage. It sets the stage for further simplification by combining like terms effectively.

Combining Like Terms

Combining like terms involves merging terms that have the same variable or variables elevated to the same power. This step reduces the number of terms in an expression, making it simpler.
Here's how to use this technique:

• Examine an expression: \(60x - 90 - 80x + 35\).
• Focus on terms with the same variable, in this case, \(x\): Combine \(60x\) with \(-80x\).
• The result for the x terms is: \[60x - 80x = -20x\]
• Next, look at constant numbers: Combine \(-90\) and \(35\).
• The result for the constants is: \[-90 + 35 = -55\]

This process, which unifies terms and cleans up the equation, is central to simplifying and finding a more concise expression.

Simplifying Expressions

Simplifying expressions involves a series of steps that aim to condense an algebraic equation into its simplest form. The goal is to present the equation as conclusively as possible, removing extra parentheses and arranging terms in an orderly manner. Let's break it down:

• Begin with complex expressions, like \(10(6x-9)-(80x-35)\).
• Use the distributive property to eliminate parentheses and distribute multiplication.
• Resulting in an expression like: \(60x - 90 - 80x + 35\).
• Combine like terms efficiently: moving step-by-step through each part of the expression.
• Simplify variables and constants separately to reach the final, clean expression: \[-20x - 55\]

Simplification helps clarify the structure of an equation, highlighting the essential quantities and relationships. It is essential in solving equations, making them easier to manage and understand. By using both the distributive property and combining like terms effectively, daunting algebra expressions transform into manageable, streamlined forms.