Problem 83 - Algebra A Combined Function

Question

Simplify each expression. See Section \(1.8 .\) \(5 x+2(x-6)\)

Step-by-Step Solution

Answer

The simplified expression is \(7x - 12\).

Step 1: Expand the Expression

We start by expanding the expression. Distribute the 2 across the expression inside the parentheses: \[ 2(x - 6) = 2 \cdot x - 2 \cdot 6 = 2x - 12 \] The expanded expression is: \[ 5x + 2x - 12 \]

Step 2: Combine Like Terms

Next, we combine like terms. We have two terms that contain \(x\):\[ 5x + 2x = 7x \]Now, the expression becomes:\[ 7x - 12 \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It's a simple but powerful tool that allows us to multiply a single term by each term inside a set of parentheses. In this exercise, we use the distributive property to get rid of the parentheses and simplify the expression, making it easier to work with.
Imagine you have the expression is:

• \(2(x - 6)\)

To apply the distributive property, you multiply the 2 by each term inside the parentheses. The steps look like this:

• First, multiply 2 by \(x\), to get \(2x\).
• Next, multiply 2 by \(-6\), which gives you \(-12\).

So, the expanded expression is \(2x - 12\). Notice how each part of the expression inside the parentheses is treated individually for multiplication. Mastering the distributive property makes it much easier to simplify and solve algebraic expressions.

Combining Like Terms

Once you've used the distributive property, the next step is often to combine like terms. This is another key concept in simplifying algebraic expressions.When we talk about "like terms," we're referring to terms that have the same variable raised to the same power. Only the coefficients (the numbers in front of the variables) can be different. For example, in the expression

• \(5x + 2x - 12\)

\(5x\) and \(2x\) are like terms because they both have the variable \(x\).
To combine them, simply add their coefficients:

• \(5x + 2x = 7x\)

And there you have it, 7x! By combining like terms, the expression becomes cleaner and more straightforward to understand. This process not only simplifies the equation but also helps to prepare it for further operations or solving if needed. Always remember to look for terms with the same variable so you can combine them easily!

Simplification

Simplification in algebra is all about breaking down complicated expressions into simpler, more manageable parts. This often involves using the distributive property and combining like terms, as we saw in the previous sections.
Let's take a look at the expression we arrived at:

• \(7x - 12\)

This expression can no longer be simplified because there are no further like terms to combine, and no operations left to perform within it.
The goal of simplification is to make an expression as straightforward as possible, often preparing it for solving. Whether you are reducing fractions, sorting out terms, or getting rid of parentheses, simplification is about creating clarity.

• It makes algebra less intimidating.
• It prepares equations for solving in later steps,
• It's a cornerstone skill in algebraic problem-solving.

Practice these steps, and you'll find that handling even the most complex initial expressions becomes second nature. Keep an eye out for these techniques whenever you're dealing with algebraic expressions.