Problem 40 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-29-53-37$$

Step-by-Step Solution

Answer

The expression simplifies to -119.

Step 1: Convert subtractions to addition of opposites

We start with the expression \(-29 - 53 - 37\).The rule for subtraction is to add the opposite. Therefore,\(-29 - 53\) becomes \(-29 + (-53)\), and,\(-37\) becomes \(+ (-37)\).So the expression \(-29 - 53 - 37\) becomes \(-29 + (-53) + (-37)\).

Step 2: Add numbers from left to right

Now that all numbers in the expression \(-29 + (-53) + (-37)\) are combined using addition, we will add from left to right:Begin with \(-29 + (-53) = -82\).Then add \(-37\):\(-82 + (-37) = -119\).Thus, the simplified result of the expression is \(-119\).

Key Concepts

Addition of Opposites

Understanding the concept of "addition of opposites" is crucial for working with negative numbers and simplifying expressions like \[-29 - 53 - 37\].
This concept means replacing subtraction with the addition of a negative number. For example, the subtraction \(-53\) is replaced by \(+(-53)\).
Changing subtractions into additions of opposites helps simplify calculations:

• It provides a consistent method applicable to different expressions, making them easier to solve.
• It eliminates confusion that might arise from dealing with mixed addition and subtraction operations.
• By focusing on addition only, it allows the use of straightforward addition rules that are commonly practiced.

This approach makes calculations more manageable and reinforces the basic arithmetic principle that subtracting a number is the same as adding its opposite. With this understanding, simplifying expressions becomes a less daunting task.

Integer Operations

Working with integers involves understanding operations with positive and negative numbers. Whether you are adding, subtracting, multiplying, or dividing, getting familiar with these rules ensures accuracy:

• Addition of two negatives: When you add two negative numbers, the result is always negative. For instance,\(-29 + (-53) = -82\).
• Addition of a positive and a negative: This involves subtracting the smaller absolute value from the larger one and taking the sign of the larger. However, in our original exercise, this rule isn't directly used because we exclusively deal with negatives.
• Subtraction as addition of opposites: Always swap the subtraction to its opposite addition to simplify calculation, as seen where \(-29 - 53\) becomes \(-29 + (-53)\).

Practicing these rules enhances your ability to simplify and solve more complex math problems. The goal is to become fluent with these operations, making it second nature when solving expressions.

Prealgebra Skills

Developing strong prealgebra skills lays the groundwork for success in higher mathematics. Simplifying expressions, especially those involving integers, is an essential skill in prealgebra. Here's why it's important:

• Building confidence: Mastering simplification techniques like changing subtraction to addition builds confidence in tackling math problems.
• Foundation for algebra: Skills developed in prealgebra pave the way for understanding more complex algebraic concepts such as solving equations and inequalities.
• Problem-solving skills: As you practice simplifying and solving expressions, your ability to tackle and solve varied problems grows stronger.
• Logical thinking: Engaging in prealgebra helps in developing logical steps and methods for approaching mathematical problems, which are valuable in everyday decision making.

By focusing on honing these skills, you become better equipped to navigate mathematical challenges with ease and clarity. Each exercise you solve adds to your knowledge and ability to handle diverse math scenarios.