Problem 70 - College Algebra

Question

simplify each algebraic expression. $$ -(-17 y) $$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression \(-(-17y)\) is \(17y\).

Step 1: Understand the Problem

The expression to simplify is \(-(-17y)\). The symbol '-' is a unary operator that changes the sign of the value or variable that follows it. Here we have two '-' operators, which represent a double negative.

Step 2: Apply the Rule of Double Negatives

According to the rule of double negatives, a double negative results in a positive. Therefore, simplifying \(-(-17y)\) results in positive \(17y\).

Step 3: Final Simplification

So, the simplified form of \(-(-17y)\) is \(17y\).

Key Concepts

Understanding Double Negatives

Encountering a double negative in an algebraic expression might seem confusing at first, but it's similar to language rules. When you use two negatives, they cancel each other out. In math, this phenomenon means that when two negative signs are together, as in \(-(-x)\), they make a positive. Simply put, subtracting a negative is the same as adding a positive. For example, in the expression \(-(-17y)\), the two negatives "undo" each other. You end up with simply \(17y\). This transformation is guided by the double negative rule, which is a fundamental principle in both arithmetic and algebra.

What's a Unary Operator?

A unary operator is an operator that affects a single operand, different from binary operators, which work with two operands. The '-' sign is a common unary operator used in algebraic expressions. It changes the sign of the number or variable it precedes. For example, if you have \(-x)\), the unary operator changes the value of \(x\) to its opposite, which is \(-x\) if \(+x\) was expected. It's important to understand this mechanism, as it frequently shows up in more complex algebraic problems. In our expression \(-(-17y)\), each negative sign acts independently at first, flipping signs of the expression. However, as they are consecutive, they produce a positive outcome due to the double negative rule.

Simplification in Algebra

Simplification in algebra involves reducing expressions to their simplest form. This means cutting down the number of operations or terms to what’s essential while retaining the original meaning. Simplifying involves working with constants, variables, and recognizing opportunities like eliminating double negatives or properly applying operators. For example, in the expression \(-(-17y)\), the simplification process involves observing the double negative and converting it to \(17y\). This removal of unnecessary negatives reduces complexity, making the expression clearer and easier to manipulate further if needed. Simplification is a critical skill in algebra, essential for solving equations and understanding the relationships between variables more swiftly.