SAT Math - How to find expression 2/3 (a² - a - 3) + 1/3 (a² + 2a + 6) is equivalent to a² ?

 

In this article, we will explore a SAT Math question that involves simplifying an algebraic expression: "Which of the following expressions is equivalent to 2/3 (a² - a - 3) + 1/3 (a² + 2a + 6)?" The correct answer is Option A) a². By the end of this exploration, you will have a clear understanding of how to simplify and identify equivalent expressions, improving your performance on the SAT Math section.

The SAT Math section evaluates your ability to work with algebraic expressions, simplifying them and identifying equivalent forms. This skill is essential for success not only on the SAT but also in various areas of mathematics and problem-solving. 

Question: Simplifying an Algebraic Expression

Question: Which of the following expressions is equivalent to 2/3 (a² - a - 3) + 1/3 (a² + 2a + 6)?

Options:

A) a²

B) a² + a

C) a² - a

D) a² - 1

Correct Answer: Option A) a²

To simplify the given expression and verify which option is equivalent, follow these steps:

Step 1: Start with the original expression:

2/3 (a² - a - 3) + 1/3 (a² + 2a + 6)

Step 2: Distribute the fractions to each term inside the parentheses:

(2/3) × a² - (2/3) × a - (2/3) × 3 + (1/3) × a² + (1/3) × 2a + (1/3) × 6

Step 3: Perform the multiplications:

(2/3) × a² = (2a²/3)

(2/3) × a = (2a/3)

(2/3) × 3 = 2

(1/3) × a² = (a²/3)

(1/3) × 2a = (2a/3)

(1/3) × 6 = 2

Step 4: Combine like terms:

(2a²/3) - (2a/3) - 2 + (a²/3) + (2a/3) + 2

Step 5: Notice that several terms cancel out:

(2a/3) and (2a/3) cancel each other.

(-2) and (+2) cancel each other.

Step 6: Simplify further:

(2a²/3) + (a²/3)

Step 7: Combine the fractions:

(2a² + a²)/3

Step 8: Add the numerators:

(3a²)/3

Step 9: Cancel the common factor of 3 in the numerator and denominator:

a²/1 = a²

Step 10: The simplified expression is a².

Step 11: The correct answer is Option A) a².

Explanation and Key Concepts:

Simplifying algebraic expressions involves distributing any coefficients to each term inside the parentheses and then combining like terms. When multiplying fractions, multiply the numerators together and the denominators together. Cancel common factors when simplifying fractions. The equivalent expression should match the original expression for all values of 'a,' which confirms that Option A) a² is the correct answer.

Simplifying algebraic expressions is a fundamental skill for the SAT Math section and beyond. Understanding how to distribute coefficients, combine like terms, and cancel common factors allows you to simplify expressions efficiently and identify equivalent forms. This skill is valuable in algebra, calculus, and various STEM fields where mathematical manipulation is essential. By mastering this skill, you'll not only excel on the SAT but also enhance your overall mathematical proficiency.



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