Problem 71 - College Algebra

Question

simplify each algebraic expression. $$ -(2 x-3 y-6) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \[-2x + 3y + 6\].

Step 1: Distribute Negative Sign

Start the exercise by distributing the negative sign to each term contained in the parentheses. This means multiplying each term within the parentheses by -1. The calculation will look as follows: \[-(2x) + -(-3y) + -(-6)\]. This calculation changes the sign of each term within the parentheses.

Step 2: Perform Distribution

Perform the aforementioned distribution to arrive at the simplified expression. This results in \[-2x + 3y + 6\].

Step 3: Final Simplified Expression

The final simplification is \[-2x + 3y + 6\]. There are no further simplifications possible as there are no like terms to combine.

Key Concepts

Distributive Property

The Distributive Property is a fundamental rule in algebra that allows you to simplify expressions. It involves distributing, or multiplying, a single term across terms inside a set of parentheses. Consider this property like handing out candy to each child in a group. Each child gets a piece, ensuring everyone is included. In algebra, when you see an expression such as \[-(2x - 3y - 6)\], you are required to distribute the negative sign to all terms inside the parentheses. It means multiplying every term inside the parentheses by \(-1\). Breaking it down, you will do the following:

• Multiply \(-1\) with \(2x\) to get \(-2x\).
• Multiply \(-1\) with \(-3y\) which changes the negative sign to positive, resulting in \(3y\).
• Lastly, \(-1\) times \(-6\) again results in a positive \(6\).

Using distributive property aids in simplifying expressions and makes them easier to work with.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression as simple as possible. It involves combining like terms, removing parentheses, and getting rid of unnecessary terms. After using the distributive property to tackle \(-(2x - 3y - 6)\), we arrive at \(-2x + 3y + 6\). Simplification is complete when there are no more parentheses, all multiplication or division has been carried out, and no like terms remain to be combined. In this expression, there are distinct terms— \(-2x\), \(3y\), and \(6\) —meaning the simplification is complete since there is nothing that can be further combined. Keeping expressions as straightforward as possible can make proceeding calculations and operations much easier to handle.

Negative Numbers

Handling negative numbers can be tricky sometimes, but with some practice, becomes quite intuitive. The key to dealing with negative numbers is understanding how their signs affect mathematical operations. When you distribute a negative number across terms within parentheses, it flips the signs of each term. Looking at the expression \(-(2x - 3y - 6)\), we use the idea that multiplying a negative by a positive results in a negative, and multiplying two negatives results in a positive.

• For the term \(-1 \times 2x = -2x\), the product is negative since one of the terms is negative.
• Similarly, \(-1 \times -3y = 3y\) results in a positive because two negatives cancel out.
• The same applies for \(-1 \times -6 = 6\).

Always pay attention to how signs change with negative numbers, as this plays a crucial role in getting the correct result.

Order of Operations

Order of operations is the set of rules that dictates the correct sequence to evaluate an expression involving multiple operations. Following the correct order ensures clarity and precision in solving algebra problems. The sequence often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the context of simplifying \(-(2x - 3y - 6)\), we first handle operations inside parentheses by distributing the negative sign according to the rules touched on previously. The expression has no exponents or further multiplications and divisions amongst distinct terms, simplifying directly into \(-2x + 3y + 6\). Remembering the order of operations ensures expressions are handled correctly from start to finish.