Problem 16 - Intermediate Algebra

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$ -37+42+18+37+(-42)-6 $$

Step-by-Step Solution

Answer

The simplified expression is 12.

Step 1: Group Positive and Negative Numbers

We start by grouping the numbers based on their signs to simplify the calculations. We have positive numbers: 42, 18, 37 and negative numbers: -37, -42, -6.

Step 2: Simplify Positive Numbers

Add all the positive numbers first: \(42 + 18 + 37 = 97\).

Step 3: Simplify Negative Numbers

Add all the negative numbers together: \(-37 + (-42) + (-6) = -85\).

Step 4: Combine the Results

Combine the sum of positive and negative numbers: \(97 + (-85) = 12\). This simplifies the entire expression.

Key Concepts

Numerical Expressions

Numerical expressions are mathematical phrases that involve numbers and operations, but they do not include an equality sign like an equation does. For instance, when you see a series of numbers combined by operations such as addition, subtraction, multiplication, or division, you are looking at a numerical expression. Simplifying these expressions involves performing the operations presented to arrive at a single numerical value.
To simplify the expression given in the exercise, you'll want to follow appropriate mathematical rules and order of operations (often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In the case of the exercise, since it involves only addition and subtraction, you can proceed straightforwardly by organizing and executing these operations methodically.

Properties of Numbers

Understanding the properties of numbers is essential when simplifying numerical expressions. These properties help make calculations cleaner and easier to manage. Let's touch on a few important ones:

• Commutative Property of Addition: This property tells us that numbers can be added in any order, and the sum will be the same. For example, in the expression \(-37+42\), rearranging the numbers to \(42 + (-37)\) does not change the result.
• Associative Property of Addition: When multiple numbers are being added together, they can be grouped in any manner without altering their sum. For instance, \((42 + 18) + 37 = 42 + (18 + 37)\).
• Identity Property of Addition: Adding zero to a number does not change its value, which is handy to remember, although we don't directly use zero in our problem.

Using these properties helps in grouping numbers effectively, making the entire simplification process much easier and reducing errors.

Addition and Subtraction

Addition and subtraction are fundamental arithmetic operations and the basis for solving our numerical expression. When performing these operations on a sequence of numbers, especially one with both positive and negative values, it's crucial to approach systematically.
Start by grouping the numbers, as was shown in the step-by-step solution. This involves separating the positive from the negative to simplify each set:

• Add the Positive Numbers: Sum all positive numbers first. In our example, \(42 + 18 + 37 = 97\).
• Add the Negative Numbers: Then, sum all negative numbers together, treating the subtraction as the addition of a negative value, \(-37 + (-42) + (-6) = -85\).
• Combine the Results: Once these are simplified, combine the results to find the sum of the original expression, \(97 + (-85) = 12\), neatly wrapping up our simplification task.

Remember, when dealing with negatives in addition and subtraction, always keep track of signs to avoid mistakes, and use parentheses where necessary to clarify the operations involved.