Problem 59 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$5(3 x+2)-4$$

Step-by-Step Solution

Answer

The simplified algebraic expression is \(15x + 6\)

Step 1: Apply the distributive property

Multiply each term inside the parentheses by 5. This yields \(5*3x+5*2=15x+10\)

Step 2: Combine like terms

After completing the first step, we are left with \(15x + 10 - 4\). Combine the constants 10 and -4 to get 6. The final simplified expression is \(15x+6\)

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term by two or more terms inside a set of parentheses. In simpler terms, it helps you to "distribute" the multiplication across the terms inside the brackets.
For example, in the expression \(5(3x+2)\), you need to multiply both terms inside the parentheses by 5.
Here's how it works step-by-step:

• Multiply \(5\) by \(3x\) to get \(15x\).
• Next, multiply \(5\) by \(2\) to get \(10\).

This gives you the expression \(15x + 10\).
By applying the distributive property correctly, you help lay the groundwork for further simplifications.

Combining Like Terms

Combining like terms is another essential skill when working with algebraic expressions. It involves simplifying expressions by merging terms that have the same variable part.
Like terms have identical variable components and can include terms like \(3x\), \(15x\), and \(7x\). Only their coefficients differ.
In our expression \(15x + 10 - 4\):

• \(15x\) is a term with a variable, while \(10\) and \(-4\) are both constant terms.
• Combine the constants: \(10 - 4\) to yield \(6\).

Thus, the resulting expression simplifies to \(15x + 6\).

The Art of Simplification

Simplification in algebra means transforming an expression into its most concise form without changing its value. The process often involves using properties like the distributive property and combining like terms.
For instance, in the initial expression \(5(3x+2) - 4\), after distributing and combining like terms, you arrive at \(15x + 6\).
This final version is much easier to work with in further calculations.Key tactics for simplification:

• Always apply the distributive property first if there are parentheses.
• Look for like terms to combine, simplifying the numerical part whenever possible.

Mastering these steps ensures clarity and accuracy in algebra.