### SAT Math - How to find expression a(b - c) - b(a + c) - c(a - b) is equivalent to - 2ac ?

## In this article, we will explore a SAT Math question that involves simplifying an algebraic expression: "Which of the following expressions is equivalent to a(b - c) - b(a + c) - c(a - b)?" The correct answer is Option D) -2ac. 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 assesses your mathematical skills, including your ability to simplify algebraic expressions and identify 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 a(b - c) - b(a + c) - c(a - b)?

### Options:

A) bc

B) 2ac

C) -2bc

D) -2ac

### Correct Answer: Option D) -2ac

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

#### Step 1: Start with the original expression:

a(b - c) - b(a + c) - c(a - b)

#### Step 2: Distribute the terms inside each set of parentheses:

(ab - ac) - (ab + bc) - (ac - bc)

#### Step 3: Perform the subtractions:

ab - ac - ab - bc - ac + bc

#### Step 4: Cancel like terms:

_ab_ - ac - _ab_ - _bc_ - ac + _bc_

#### Step 5: Simplify further:

- ac - bc

#### Step 6: The simplified expression is -2ac + 2bc.

#### Step 7: The correct answer is Option D) -2ac.

### Explanation and Key Concepts:

Simplifying algebraic expressions involves distributing terms inside parentheses, combining like terms, and simplifying further. When subtracting terms, remember to apply the subtraction to each term within the parentheses. The equivalent expression should match the original expression for all values of 'a,' 'b,' and 'c,' which confirms that Option D) -2ac is the correct answer.

Simplifying algebraic expressions is a fundamental skill for the SAT Math section and beyond. Understanding how to distribute terms, combine like terms, and factor out 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.