Problem 61 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 2(x-6)+5 $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given algebraic expression is \(2x - 7\).

Step 1: Expand the parenthesis

To expand the parenthesis, we need to distribute the outside term (in this case, the 2) to all the terms inside the parenthesis. So, multiply 2 by each term within \((x-6)\). This gives: $$ 2(x-6) + 5 = 2x - 12 + 5 $$

Step 2: Combine like terms

Now that we've expanded the parenthesis, we can combine any like terms. In this case, we have the constant terms -12 and 5 that can be combined. This results in: $$ 2x - 12 + 5 = 2x - 7 $$ Now the expression has been simplified to \(2x - 7\).

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions means making them as straightforward as possible without changing their value. When you simplify an expression, you aim to reduce it to its most concise form. In the example given, the expression \(2(x-6)+5\) was simplified. The process involved expanding the terms within the parentheses and then combining any like terms. The goal is to end up with an expression that is easier to work with in calculations.

• Every step in simplification reduces the complexity of the original expression.
• A simplified expression will be easier and quicker to work with, whether you are solving equations or substituting values.
• Simplification can involve various operations such as addition, subtraction, multiplication, and division.

The final simplified expression in the example was \(2x - 7\), which is much shorter and cleaner.

Distributive Property

The distributive property is an essential tool in algebra that helps you remove parentheses by distributing a factor across terms inside the parentheses. In other words, it lets you multiply each term inside the parentheses by the term outside. This property is crucial when working with expressions that contain parentheses and ensures you apply multiplication correctly across all terms.

• In the expression \(2(x-6)+5\), the number 2 outside the parentheses was distributed to both \(x\) and \(-6\).
• This expanded the expression to \(2x - 12\), thus removing the parentheses.
• Using the distributive property correctly helps prevent mistakes when simplifying expressions.

Applying the distributive property can ease the process of simplification by clearly laying out each component of the expression.

Combining Like Terms

Combining like terms is a fundamental skill in algebra that involves adding or subtracting coefficients of terms that have the same variable part. This step often follows the use of the distributive property to further simplify the expression. Like terms have the exact same variable and exponent.

• In the simplified form \(2x - 12 + 5\), the terms \(-12\) and \(5\) are considered to be like terms because they are both constant terms.
• Combining \(-12\) and \(5\) by performing the operation \(-12 + 5\) gives \(-7\), resulting in \(2x - 7\).
• This step ensures the expression contains the minimum number of terms possible, making it easier to interpret and use in calculations.

By combining like terms, expressions become streamlined, reducing potential errors in further algebraic operations.