Problem 93 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$-(2 y-5)$$

Step-by-Step Solution

Answer

The simplified form of the expression \( -(2y-5) \) is \( 5 - 2y \)

Step 1: Distribute -1 to each term inside the parentheses

To simplify the algebraic expression, distribute -1 (or the negative sign) to each term inside the parentheses. In other words, multiply each term inside the parentheses by -1.\n\So, \( -(2y - 5) \) becomes \( -1 * 2y + -1 * -5 \)

Step 2: Perform each multiplication

Perform each multiplication separately. So, we get \( -2y + 5 \)

Step 3: Rewrite the result

Rewrite the result in a more standard format by reversing the two terms and changing the plus understood as addition to a minus sign understood as subtraction.\n\So, \( -2y + 5 \) becomes \( 5 - 2y \)

Key Concepts

Distributive Property

Understanding the distributive property is essential when working with algebraic expressions. This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then summing the products. In other words, if you have an expression like \( a(b + c) \), you can 'distribute' the \( a \) across the \( b \) and the \( c \), resulting in \( ab + ac \).

Let's apply this to our problem. The expression \( -(2y - 5) \) involves a negative sign outside parentheses, which is the same as multiplying by \( -1 \). When we distribute the \( -1 \), we multiply it by each term inside the parentheses, leading to \( -1 \times 2y + -1 \times -5 \). The distributive property shows its power here, as it allows us to simplify complex algebraic expressions by breaking them down into simpler parts.

Multiplying Integers

Multiplying integers might seem straightforward, but it can get tricky with positive and negative numbers. Remembering the basic rules helps: a positive times a positive gives a positive, a negative times a negative gives a positive, and a positive times a negative gives a negative. It's like mixing colors; certain combinations give predictably consistent results.

In our exercise, we multiply integers inside an algebraic expression. We have \( -1 \times 2y \), which gives us \( -2y \), because we're combining a negative and a positive. Meanwhile, \( -1 \times -5 \) gives us positive \( 5 \), because a negative times a negative is a positive. Successfully managing the signs during multiplication is key to correctly simplifying algebraic expressions.

Algebraic Manipulation

Algebraic manipulation involves rearranging terms and factors within an expression to make it simpler or to bring it into a form that's easier to work with. One aspect of it is understanding the 'order of operations,' which ensures that operations are performed in a systematic way - parentheses first, then exponents, followed by multiplication and division, and finally addition and subtraction.

After distributing and multiplying in our example, we're left with \( -2y + 5 \). This is our simplified expression, but we can manipulate it further by reversing the terms for aesthetic or conventional purposes, ending up with \( 5 - 2y \). Such steps are sometimes taken for a clearer presentation, to match a standard form, or to set up an equation for solving. Tuning your skills in algebraic manipulation is fundamental to working through algebra problems efficiently.