Problem 20 - Elementary Algebra

Question

Simplify. $$ 6 x 2-4 x+7 x 2-3 x $$

Step-by-Step Solution

Answer

The simplified expression is \(13x^2 - 7x\).

Step 1: Identify Like Terms

In the expression \(6x^2 - 4x + 7x^2 - 3x\), identify terms with the same variables and exponents. Here, \(6x^2\) and \(7x^2\) are like terms, and \(-4x\) and \(-3x\) are like terms.

Step 2: Combine Like Terms for \(x^2\)

Add the like terms \(6x^2\) and \(7x^2\) to get a single term. This gives \((6 + 7)x^2 = 13x^2\).

Step 3: Combine Like Terms for \(x\)

Add the like terms \(-4x\) and \(-3x\) to get a single term. This gives \((-4 - 3)x = -7x\).

Step 4: Write the Simplified Expression

Combine the results from Steps 2 and 3 to get the simplified expression: \(13x^2 - 7x\).

Key Concepts

Understanding Like Terms

Like terms are an essential concept in simplifying polynomials. They are terms that contain the exact same variables raised to the same powers.
For example, in the expression \(6x^2 - 4x + 7x^2 - 3x\), \(6x^2\) and \(7x^2\) are like terms because they both involve \(x\) raised to the power of 2. Similarly, \(-4x\) and \(-3x\) are like terms as they both involve \(x\) to the power of 1.
It's crucial to only combine terms that are like terms, as combining unlike terms can lead to incorrect simplification.

• Identify terms with the same base and exponent as like terms.
• Remember that coefficients can be different; only the variables and their exponents must match.

How to Combine Terms

Once like terms have been identified, the next step is to combine them. Combining terms involves adding or subtracting their coefficients.
For our example expression \(6x^2 + 7x^2\), we add the coefficients (6 and 7) to get \(13x^2\). The variable part \(x^2\) remains unchanged because we only focus on the coefficients.
This step helps reduce the expression to fewer terms, making it simpler and easier to understand.

• Add the coefficients of like terms.
• Keep the variable part the same; don’t change its power.
• Apply the same principle to negative terms: for \(-4x - 3x\), combine to get \(-7x\).

Breaking Down Algebraic Expressions

Algebraic expressions are combinations of numbers and variables connected by basic operations like addition and subtraction.
Understanding algebraic expressions is crucial because it forms the foundation of algebra, enabling you to simplify, solve, and manipulate equations.
In the expression \(6x^2 - 4x + 7x^2 - 3x\), we see several parts:

• Terms, which are parts separated by plus or minus signs, like \(6x^2\), \(-4x\), \(7x^2\), and \(-3x\).
• Coefficients, which are the numerical parts in front of the variables (e.g., 6, -4, 7, -3).
• Variables, which are symbols that represent numbers (\(x\) in this case).

Recognizing each component of an algebraic expression helps in simplifying it and solving related problems more effectively.