Problem 91 - Introductory and Intermediate Algebra for College Students 4th

Question

Exercises \(91-93\) will help you prepare for the material covered in the next section. Simplify: \(13-3(x+2)\)

Step-by-Step Solution

Answer

The simplified form of the expression is \(7 - 3x\).

Step 1: Apply Distributive Property

First, apply the distributive property which means to multiply the -3 with each term inside the parentheses: \(13 - 3*(x + 2) = 13 - 3x - 6\)

Step 2: Combine Like Terms

Next, combine like terms, which are constants in this scenario, to fully simplify the expression: \(13 - 6 = 7\), resulting in the simplified expression of \(7 - 3x\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept used in algebra to simplify expressions and solve equations. This property basically means that you multiply a single term by each term inside a set of parentheses. This process helps in breaking down expressions into simpler parts.
A great way to think about it is, if you have something like \( a(b + c) \), the distributive property lets you rewrite this as \( ab + ac \).
In the context of our exercise, we need to apply the distributive property to \( -3(x + 2) \). Here, you take \(-3\) and multiply it both by \( x \) and by \( 2 \).
This will give you:

• The first term, \(-3 \times x = -3x\)
• And the second term, \(-3 \times 2 = -6\).

After applying the distributive property, our expression becomes \( 13 - 3x - 6 \). Notice how each term inside the parentheses got multiplied individually.

Combining Like Terms

Once you've used the distributive property, the next step often involves combining like terms. This simplifies the expression further, making it easier to understand or solve. Like terms are terms that have the same variable raised to the same power.
For instance, in the expression we ended up with, \( 13 - 3x - 6 \), the terms \( 13 \) and \( -6 \) are like terms because they are both constants (they have no variable).
To combine them, you simply add or subtract the numbers just as you would in regular arithmetic.
Adding those like terms:

• \( 13 - 6 = 7 \)

This results in a simplified, combined expression of \( 7 - 3x \). The variable term \(-3x\) remains unchanged, as there are no other \(x\) terms to combine it with. This step condenses the expression to its simplest form without altering its value.

Simplifying Expressions

Simplifying expressions is about making an algebraic expression easier to understand or solve while keeping its value the same. It's a cornerstone in algebra, pivotal for solving equations or even just interpreting a mathematical problem accurately.
The process involves several steps:

• First, we apply properties such as distributive, associative, or commutative, if needed.
• Then, we carefully combine any like terms.

During simplification, it’s important to be precise with the mathematical operations you perform.
For our specific case, starting from \( 13 - 3(x + 2) \), after applying the distributive property and combining like terms, we reached the simpler form of \( 7 - 3x \).
This new expression, \( 7 - 3x \), is more streamlined and makes it easier for us to understand the relationship between the terms or proceed with further algebraic manipulations. Simplifying ensures that unnecessary complexity is removed, allowing a clearer view of the expression's true meaning.
Remember, simplification does not change the value; it makes the expression cleaner and more practical for usage.