Problem 60 - Elementary Algebra

Question

Simplify. $$ 3 x 2+24 x 2-2 x-8-12 x-4 $$

Step-by-Step Solution

Answer

The simplified expression is \(13x - 8\).

Step 1: Distribute Multiplication

We start by distributing the multiplication in the expression. The expression is given in a format that looks as though it may contain multiplication, but actually it contains addition and subtraction terms directly. Let's reorganize it in an expanded form:\[ 3x + 2 + 24x + 2 - 2x - 8 - 12x - 4 \]

Step 2: Combine Like Terms

Next, we identify and combine like terms. Variables that are the same, such as all terms containing \(x\), are combined together while constants are added together separately:- Combine \(3x\), \(24x\), and \(-2x\), \(-12x\): \[ 3x + 24x - 2x - 12x = 13x \]- Combine constants \(2\), \(2\), \(-8\), and \(-4\): \[ 2 + 2 - 8 - 4 = -8 \]

Step 3: Final Simplified Form

Finally, we combine the simplified variable terms and constants to write the expression in its completely simplified form:\[ 13x - 8 \]

Key Concepts

Combining Like Terms

In algebra, "like terms" refer to terms that have the same variable raised to the same power. Combining like terms is an essential skill in simplifying expressions. It involves adding or subtracting coefficients of these terms to consolidate them into a single term.Imagine you're organizing a box of letters. You group all the 'x' letters together and all the constants together. For the expression we have, that's exactly what combining like terms is all about:

• Look at the terms with the variable \(x\): \(3x\), \(24x\), \(-2x\), and \(-12x\).
• Add the coefficients: \(3 + 24 - 2 - 12 = 13\).
• Combine these to form \(13x\).

Similarly, you handle the constants separately by adding them up. This process is key to simplifying algebraic expressions efficiently.

Distributive Property

The distributive property is a fundamental property of multiplication over addition or subtraction. However, in this exercise, the use of the distributive property was more about reorganizing the expression since no explicit multiplication was present initially.It allows us to multiply a single term by each term within a parenthesis. For example, for \(a(b + c)\), it distributes to become \(ab + ac\).In this exercise:

• The expression was first visually reformatted to clearly show all terms: \(3x + 2 + 24x + 2 - 2x - 8 - 12x - 4\).
• Understanding the structure is essential even if we don't perform direct distribution; we ensure the order of operations is respected and prepare for combining like terms.

Identifying parts of an expression that resemble distributive forms, even without actual distribution, can reinforce comprehension of how terms aggregate.

Variables and Constants

In algebra, expressions are made up of variables and constants, which are the building blocks you manipulate.Variables are symbols (often letters like \(x\)) that represent unknown values. They're dynamic and can change based on conditions of a problem. In our exercise, \(x\) is a variable.Constants are fixed numbers without variables attached. They have set values irrespective of what you do to the variables. Here, numbers like 2, -8, and -4 are constants in the expression.Understanding these components is crucial as:

• Combining like terms depends on correctly identifying which parts are variables and which are constants.
• This basic distinction enables effective organization and simplification of expressions.

This clarity forms the foundation for working with more complex algebraic expressions.