Problem 97 - Algebra A Combined Function

Question

Each calculation below is incorrect. Find the error and correct it. $$ 9-(-7) \stackrel{?}{=} 2 $$

Step-by-Step Solution

Answer

9 - (-7) = 16.

Step 1: Identify the Operation Error

The given expression is \(9 - (-7)\), which should be simplified correctly. The mistake here is related to the handling of double negatives. When subtracting a negative number, the negative signs cancel out, transforming the expression.

Step 2: Correct the Double Negative

The expression \(9 - (-7)\) involves subtracting a negative number. Use the rule of 'minus a negative equals a positive':\[9 - (-7) = 9 + 7\]

Step 3: Perform the Correct Calculation

Now that the operation is accurately set as an addition, compute the sum:\[9 + 7 = 16\]

Step 4: Verify the Answer

Confirm that the corrected calculation \(9 + 7 = 16\) is consistent. The original result of 2 clearly should be 16 according to the correct application of subtraction of negatives.

Key Concepts

Operation Errors

Operation errors are common when performing arithmetic calculations, especially with negative numbers. These mistakes can arise due to misunderstandings of how number operations interact, particularly subtraction and the handling of negative numbers.
When you see an expression like \(9 - (-7)\), it’s easy to misinterpret the subtraction of a negative number. A frequent error is to treat this as a simple subtraction (\(9 - 7 = 2 \)), but this operation doesn’t take the double negative into account.

• Always remember that subtraction is not the same as addition.
• Negative signs must be carefully considered in each arithmetic operation.

Being mindful of these potential missteps is crucial to avoiding operation errors.

Double Negatives

Double negatives can often be tricky because they involve two negative signs. In mathematics, particularly in operations like subtraction, understanding how these negatives interact is key.
When you see two negatives next to each other, like in \(9 - (-7)\), they effectively cancel each other out, transforming the operation into addition.

• This is due to the rule that 'minus a negative equals a positive'.
• So, \(9 - (-7) = 9 + 7\).

This rule simplifies the computation, switching the perspective from removal to addition, which most find easier to manage.

Arithmetic Simplification

Arithmetic simplification is a process that helps clarify and streamline mathematical expressions. In our example, after identifying the double negative, it is crucial to resolve it into a simpler form.
By transforming \(9 - (-7)\) into \(9 + 7\), we simplify the operation significantly.

• Simplified expressions are easier to calculate mentally.
• This step reduces the chance of errors and ensures you get the correct answer.

The final step is straightforward: compute the sum to find the correct answer, \(16\). This process encourages a clear path from problem to solution, enhancing arithmetic accuracy and confidence.