Problem 15 - Elementary Algebra

Question

Add and subtract. $$ -30+20-8-(-18) $$

Step-by-Step Solution

Answer

0

Step 1: Simplify the Negation

The expression to solve is: \[-30 + 20 - 8 - (-18)\]First, simplify the subtraction of a negative number. Subtracting \(-(-18)\) is equivalent to adding 18. So, modify the expression to:\[-30 + 20 - 8 + 18\]

Step 2: Add and Subtract from Left to Right

Now simplify the expression by performing addition and subtraction from left to right.1. \(-30 + 20 = -10\)Modify the expression:\[-10 - 8 + 18\]2. \(-10 - 8 = -18\)Modify the expression:\[-18 + 18\]

Step 3: Final Calculation

Finally, evaluate the last part of the expression:\(-18 + 18 = 0\)

Key Concepts

Simplifying Expressions

When working on algebra problems, simplifying expressions is crucial for finding a solution efficiently. In our given problem, the expression is \[-30 + 20 - 8 - (-18)\]. To simplify this, you need to identify and manage any operations that can be made easier.

This expression contains a subtraction of a negative number. According to the rules of arithmetic, subtracting a negative is the same as adding its positive counterpart. Therefore, you can rewrite \(-(-18)\) as \(+18\). This transforms the original expression into a more manageable form:\[-30 + 20 - 8 + 18\]. Without simplification, solving would be unnecessarily complex.

• Rewrite complicated constructs: like double negatives.
• Look for combinations: adding and subtracting terms that can immediately cancel out or simplify, like the \(-(-18)\) identified.

Addition and Subtraction

The order of operations is important when evaluating expressions that involve both addition and subtraction. When you face such problems, it is crucial to move from left to right.

The modified version of our expression: \[-30 + 20 - 8 + 18\] can be solved by simply working through each step sequentially.

• First, \(-30 + 20 = -10\). Now the expression is \[-10 - 8 + 18\].
• Next, handle the subtraction: \(-10 - 8 = -18\). Now, you have \[-18 + 18\].
• Finally, perform the addition: \(-18 + 18 = 0\).

Approaching in a left-to-right manner helps ensure that you're not violating any arithmetic rules, keeping your solution both accurate and straightforward.

Negative Numbers

Negative numbers can often lead to confusion, especially when they appear in conjunctions such as subtraction of negatives. Understanding the nature of negative values helps in streamlining calculations and avoiding common pitfalls.

For instance, subtracting a negative number—like the instance of \(-(-18)\)—means the same as adding a positive one, which in this case is \(+18\).

• Think of negative numbers as movements on a number line: subtracting a negative means moving forward.
• When adding or subtracting with negative numbers, always carefully consider their signs. Signs dictate the direction of your calculations on that number line.

By understanding these aspects of negative numbers, problems will seem less complex and solutions easier to find.