Problem 43 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ -(a-b), \text { if } a=0 \text { and } b=-6 $$

Step-by-Step Solution

Answer

The value of the expression is \(-6\).

Step 1: Substitute the given values

First, we need to substitute the given values of \(a\) and \(b\) into the expression \(-(a-b)\). We know that \(a=0\) and \(b=-6\). So the expression becomes \(-(0 - (-6))\).

Step 2: Simplify the expression inside the parentheses

Now, simplify the expression inside the parentheses. The expression \(0 - (-6)\) simplifies to \(0 + 6\) because subtracting a negative is the same as adding a positive. Thus, we have \(6\).

Step 3: Apply the negative sign

Apply the negative sign outside the parentheses to the simplified result. So, we have \(-6\).

Key Concepts

Substitution

In basic algebra, substitution is a foundational skill that involves replacing variables with their given numerical values. This process is crucial as it allows us to evaluate expressions where the actual numbers are specified later. In the context of the given exercise, we are given the expression \(-(a-b)\) and need to substitute the values for \(a\) and \(b\).

Here’s how it works:

• Start by identifying the values specified: \(a=0\) and \(b=-6\).
• Next, replace the variable \(a\) with \(0\) and \(b\) with \(-6\) in the expression. So, \(-(a-b)\) becomes \(-(0 - (-6))\).

By correctly substituting these numbers, we convert the algebraic expression into a numerical one that can be simplified. Substitution simplifies the complex expression evaluation and is an elementary step towards solving algebraic problems.

Simplification

Simplification in mathematics refers to the process of reducing a mathematical expression to its most concise form. It involves combining like terms, reducing fractions, or resolving expressions involving operations like addition or subtraction.

Within the given problem, after substitution, we encounter the expression \(0 - (-6)\). The rule when simplifying such an expression is to remember that subtracting a negative number is equivalent to adding its positive counterpart. So:

• The expression \(0 - (-6)\) simplifies to \(0 + 6\), which results in \(6\).

Conclusively, simplification allows us to unlock the straightforward mathematical truth hidden within a more complex looking expression. It is a skill that requires familiarity with basic arithmetic operations, and the rules governing them, especially when dealing with negative numbers.

Negative Numbers

Working with negative numbers can initially appear daunting, but understanding a few core concepts can simplify the task. Negative numbers are values less than zero, often representing opposite values or directions.

A crucial rule to understand here is when subtracting negative numbers:

• Subtracting a negative is akin to adding the corresponding positive. For instance, \(0 - (-6)\) is the same as \(0 + 6\).

In the final step of our exercise, a negative sign is placed in front of \(6\), resulting in \(-6\). This transformation emphasizes that the operations conducted with negative numbers can significantly affect the outcome. Grasping these rules not only aid in solving algebraic expressions but also sharpen your arithmetic skills in everyday calculations with negative values.