Problem 38 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-7-3-6$$

Step-by-Step Solution

Answer

The simplified result is -16.

Step 1: Convert Subtraction to Addition of Opposites

To convert subtractions to additions, treat every subtraction as the addition of a negative. Thus, the expression \[-7 - 3 - 6\]becomes \[-7 + (-3) + (-6)\].

Step 2: Add Left to Right

Start by adding the first two numbers:\(-7 + (-3) = -10\).Next, add the result to the third number:\(-10 + (-6) = -16\).

Key Concepts

Addition of Opposites

Understanding the addition of opposites is a key concept when working with integer operations. At first glance, the idea may sound confusing, but it's quite simple. When you encounter subtraction in an expression, think of it as adding a negative number.

• This means changing something like \(a - b\) into \(a + (-b)\).
• In the context of our given exercise, \(-7 - 3 - 6\) can be seen as \(-7 + (-3) + (-6)\).

This method is useful because it allows you to consistently use the same operation, which is addition, throughout your calculations.
This way, you reduce the likelihood of errors that can occur when alternately switching between addition and subtraction.
Mastering this concept will simplify your calculations, as it lets you think of everything as one big addition problem.

Subtraction in Algebra

Subtraction in algebra can sometimes be challenging, especially when it's mixed with negative numbers. But, as we transform every subtraction into an addition of opposites, it becomes more manageable.
Let's revisit our exercise: \(-7 - 3 - 6\). By turning this into \(-7 + (-3) + (-6)\), you apply a consistent rule.
This method transforms the way subtraction is perceived:

• When subtracting a positive number, you're essentially moving left on the number line.
• Subtracting a negative number would mean moving right, closer to zero, because the two negatives create a positive.

This technique not only helps to solve problems more efficiently but also builds a deeper understanding of how numbers interact with one another.

Simplifying Expressions

Simplifying expressions is the process of making a mathematical phrase as simple as possible. In our exercise, the goal was to simplify \(-7 - 3 - 6\) to its reduced form.
After converting all subtractions to additions, the next step is to add them left to right in sequence.
Firstly, combine the first two numbers: \(-7 + (-3) = -10\). Then, take that intermediate result and add the next number: \(-10 + (-6) = -16\).
By consistently applying operations in a left-to-right manner, you maintain clarity in problem-solving.
This way of simplifying ensures that expressions are handled in a systematic, organized way, reducing mistakes and enhancing accuracy. Applying these steps daily will sharpen your skills and make algebra more approachable and enjoyable.