Problem 48 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 3(x-5)-2 $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(3x - 17\).

Step 1: Apply the Distributive Property

Firstly, apply the distributive property to the term within the parenthesis by multiplying 3 with each term inside the parenthesis. This gives: \[3x - 15 - 2\]. The distributive property, a cornerstone of arithmetic operations, allows you to multiply a single term and two or more terms inside a set of parentheses.

Step 2: Combine like terms

Now, proceed to combine all the like terms in the expression. In this case, the terms -15 and -2 are similar because they are both constants without variables. When these two are combined, you will get: \[3x - 17\]. Combining like terms is a way to simplify mathematical expressions.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions. It involves multiplying a single term by each term inside a set of parentheses. For example, in the expression \(3(x-5)\), the distributive property tells us to multiply both \(x\) and \(-5\) by 3. This results in \(3x - 15\).
This property can be summarized by the formula:

• \(a(b + c) = ab + ac\)

Using the distributive property makes complex equations easier to manage, especially when dealing with larger expressions.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. It involves adding or subtracting terms that have the same variable or are constants. In the expression \(3x - 15 - 2\), the like terms \(-15\) and \(-2\) are both constants.
This step gives us a simpler expression:

• \(3x - 17\)

This process helps in reducing the complexity of expressions and makes further calculations easier.

Expressions and Equations

Expressions and equations form the backbone of algebra. An expression is a combination of numbers, variables, and operations without an equality sign. For instance, \(3(x-5)-2\) is an expression.
Equations, on the other hand, have an equality sign and show a relationship between two expressions. Understanding the difference allows you to approach problems with the right strategies to simplify and solve them.

• Expressions: Evaluate or simplify
• Equations: Solve for unknowns

Arithmetic Operations

Arithmetic operations, including addition, subtraction, multiplication, and division, are the basic building blocks in algebra and mathematics as a whole. When simplifying expressions, you'll often use these operations to manipulate terms.
In the example \(3(x - 5) - 2\), multiplication is used initially with the distributive property, followed by addition to combine like terms.
Mastering these operations is essential for tackling algebraic problems effectively:

• Addition: Combine quantities
• Subtraction: Find differences
• Multiplication: Scale quantities
• Division: Divide quantities