Problem 100 - Algebra A Combined Function

Question

Each calculation below is incorrect. Find the error and correct it. $$ -3-(-10) \stackrel{?}{=}-13 $$

Step-by-Step Solution

Answer

The correct result is 7, not -13.

Step 1: Identify the Operation

Look at the given expression \(-3 - (-10)\). Here, subtraction of a negative number is taking place. To simplify, remember that subtracting a negative number is the same as adding its positive counterpart.

Step 2: Simplify the Expression

Replace the subtraction of a negative number with addition: \(-3 - (-10)\) becomes \(-3 + 10\). Simplifying the expression helps in identifying the mistake in the original calculation.

Step 3: Perform the Addition

Calculate \(-3 + 10\). Start by recognizing that adding a positive number to a negative number means moving towards the positive side on the number line. Thus, \(-3 + 10 = 7\).

Step 4: Compare with Given Answer

The initial claim was \(-3 - (-10) = -13\), but we found that the correct result is \(-3 + 10 = 7\). Clearly, the calculation of \(-13\) was incorrect.

Key Concepts

Subtraction of Negative Numbers

When we encounter subtraction involving negative numbers, it can initially seem perplexing. Subtraction of a negative number is, in fact, equivalent to addition. This concept stems from the rules of integer arithmetic: subtracting a negative is the same as adding the positive equivalent.

In the exercise provided, the expression \(-3 - (-10)\) looked as though it suggested a reduction, but in reality, you add the inverse.

• Recognize that double negatives become positive.
• The conversion is \(- (-10)\ = +10\), hence \(-3 + 10\).

Remember, subtraction of a negative number often trips up learners because it appears counterintuitive. However, with practice, this can become second nature.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves breaking down an equation to its basic form. This helps make complex calculations easier and clearer. In the context of the exercise, simplifying was crucial because it revealed the true nature of the arithmetic operation we're performing.

For example, when given \(-3 - (-10)\), the simplification process involves:

• Lifting out any double negatives.
• Recognizing and converting \(- (-10)\) into \(+10\).
• Rewriting the entire expression as \(-3 + 10\).

The simplified form makes calculations straightforward and avoids common mistakes. Simplifying helps in visualizing the transformation of numbers, leading to accurate results.

Number Line Operations

Using a number line can significantly aid in understanding arithmetic operations involving positive and negative numbers. The number line serves as a visual guide to accurately perform operations like addition and subtraction.

In the example \(-3 + 10\), visualizing on a number line helps:

• Start at \(-3\) on the number line.
• Move 10 steps to the right (since we are adding positive 10).
• Land on \(7\), giving you \(-3 + 10 = 7\).

This visual approach clarifies why adding a positive number shifts the position to the right, increasing value. Number line operations give tangible insight into abstract arithmetic concepts, anchoring your understanding of integers.