Problem 99 - College Algebra

Question

Write each algebraic expression without parentheses. \(-(2 x-3 y-6)\)

Step-by-Step Solution

Answer

The algebraic expression without the parentheses is \[-2x + 3y + 6\].

Step 1: Distribute the -1

Multiply each term inside the parentheses by -1. This yields: \[-1 * 2x - (-1 * 3y) - (-1* 6)\].

Step 2: Perform the multiplication

Perform each multiplication to obtain: \[-2x + 3y + 6\].

Step 3: Final Expression

The final expression without the parentheses is \[-2x + 3y + 6\].

Key Concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and form the language through which we express mathematical ideas using numbers, variables, and operations. An algebraic expression may include constants, coefficients, variables like x and y, and arithmetic operations such as addition, subtraction, multiplication, and division. For instance, in the expression 2x - 3y + 4, 2 is a coefficient, x and y are variables, and the numbers and variables are combined using addition and subtraction.

Understanding how these expressions work is fundamental in solving algebraic problems. They can represent real-world quantities in equations that can be solved or manipulated to answer questions. For example, an expression like 5x + 2 can represent the total cost if each item costs 5 dollars and there's an additional fee of 2 dollars. Grasping these concepts allows students to translate real scenarios into mathematical terms and then solve for unknowns.

Removing Parentheses in Algebra

The process of removing parentheses in algebra involves applying the distributive property, which allows you to simplify expressions so that they no longer contain any parentheses. It's crucial because it helps in simplifying algebraic expressions and solving equations more easily.

For example, in the exercise provided, to remove the parentheses from the expression \( - (2x - 3y - 6) \) we distribute the negative sign, which is equivalent to multiplying each term by -1. After distributing, we obtain \( -2x + 3y + 6 \) where each term inside the parentheses has been affected by the negative sign, essentially flipping the sign of each. Mastering this technique is a fundamental skill in algebra that every student needs, as it's a stepping-stone to solving more complex equations.

Simplifying Algebraic Expressions

Simplifying an algebraic expression means to make it as concise as possible, typically by combining like terms and eliminating parentheses. This process often involves the use of the distributive property and other algebraic rules.

In our example, after removing parentheses using the distributive property, we went from \( - (2x - 3y - 6) \) to \( -2x + 3y + 6 \). The expression is now simplified because it's written in its most straightforward form, free from parentheses and with all like terms combined if there were any. This step is essential for further manipulation of the expression, such as solving for a variable or substituting values to evaluate the expression. Simplification is a common practice in algebra that allows for clearer understanding and easier computation.