Problem 39 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ -2(3 x-4)+7 x-6 $$

Step-by-Step Solution

Answer

The expression simplifies to \(x + 2\).

Step 1: Apply the distributive property

To remove the parentheses, you need to distribute the -2 across the terms inside the parentheses. This means multiplying -2 by each term inside the parentheses: \(-2(3x - 4) = -2 \times 3x + (-2) \times (-4) \). Calculate it to get \(-6x + 8\).

Step 2: Combine like terms

Now, substitute back the expression from Step 1 into the original expression: \(-6x + 8 + 7x - 6\). Next, combine the like terms: Add the x terms: \(-6x + 7x = x\). Add the constant terms: \(8 - 6 = 2\).

Step 3: Write the final simplified expression

Combine the results from Step 2 to form the final simplified expression: \(x + 2\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to expand expressions. It's particularly useful when dealing with parentheses. In the exercise given, you start by distributing

• Multiply the number outside the parentheses by each term inside it.
• In our problem, the number was \(-2\) and the terms inside were \(3x\) and \(-4\).

Let's look at it closely. Applying the distributive property, you perform these calculations:
\(-2 \times 3x = -6x\)
\(-2 \times -4 = 8\)
With these simple multiplications, you transform the expression \(-2(3x - 4)\) into \(-6x + 8\). This step effectively removes the parentheses and sets the stage for further simplification. Understanding this process is key to handling more complex algebraic expressions.

Combining Like Terms

In algebra, combining like terms entails merging the terms that have the same variable raised to the same power. This process helps simplify expressions into a more manageable form. Once we've used the distributive property, our expression turned into \(-6x + 8 + 7x - 6\).

• Notice that the terms \(-6x\) and \(7x\) both have the variable \(x\).
• The numbers \(8\) and \(-6\) are constants, meaning they're just plain numbers without variables.

Combining the \(x\) terms gives you:
\(-6x + 7x = x\)
For the constants, combining gives:
\(8 - 6 = 2\)
It's like gathering similar items together to make them easier to count. Only terms exactly alike get added or subtracted, keeping the expression neat and tidy.

Parentheses Removal

Removing parentheses is essential in simplifying expressions. It makes everything more straightforward to work with. Initially, parentheses can seem like barriers, but by using the distributive property, we can easily break these barriers down.

• In our expression, we removed parentheses by distributing \(-2\) across \(3x - 4\).
• The result is a simpler, linear expression without extra layers.

This removal means you can focus on combining terms directly without any hindrance. Mastery of this step often makes subsequent processes, like term combination, more intuitive. Understanding how to handle parentheses ensures you can tackle more complex algebraic problems with confidence.

Simplification

Simplification of algebraic expressions means converting complex expressions into their simplest form. This often involves distributing, combining like terms, and organizing the expression neatly. In our given problem, after distributing and combining like terms, we were left with a clear and concise expression:
\(x + 2\).
Simplification has several benefits:

• Makes expressions more understandable.
• Facilitates further algebraic operations.
• Helps in solving equations more easily.

It’s like cleaning up your workspace, so it's easier to find things and continue working efficiently. Once everything is streamlined, it saves time and effort, especially in later stages of mathematical problem-solving.