Problem 23 - 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. $$ 14-12-21-14+17-18+19-32 $$

Step-by-Step Solution

Answer

The simplified expression is -47.

Step 1: Identify Like Terms

Group the positive and negative terms separately: Positive terms are \(14, 17, 19\) and negative terms are \(-12, -21, -14, -18, -32\).

Step 2: Simplify Positive Terms

Add all the positive terms together: \(14 + 17 + 19 = 50\).

Step 3: Simplify Negative Terms

Add all the negative terms together while keeping them negative: \(-12 - 21 - 14 - 18 - 32\). First, add in pairs: \(-12 - 21 = -33\), \(-14 - 18 = -32\), then \(-33 - 32 = -65\). Lastly, add remaining: \(-65 - 32 = -97\).

Step 4: Combine Simplified Terms

Subtract the total negative terms from the positive sum: \(50 - 97 = -47\).

Step 5: Check for Accuracy

Ensure calculations follow the order of operations and properties correctly by re-evaluating each group sum and final subtraction.

Key Concepts

Numerical Expressions

Numerical expressions are combinations of numbers and operations like addition, subtraction, multiplication, and division. In simple terms, it’s a way to express calculations without equal signs or variables. For example, the expression \(14 - 12 - 21 - 14 + 17 - 18 + 19 - 32\) is a numerical expression.
Navigating numerical expressions involves evaluating the operations in the correct sequence to achieve the correct result. The goal is to simplify them so they are easier to work with, typically resulting in a single number at the end. This requires carefully carrying out all the operations included while respecting their order to ensure accuracy.

Like Terms

Like terms play an important role in simplifying expressions. In the context of our example, identifying like terms involves spotting which numbers or expressions can be grouped and simplified together.
For numerical expressions, because we aren't using variables, we focus on grouping by their signs (positive or negative). Here, the positive terms \(14, 17, 19\) are grouped separately from the negative terms \(-12, -21, -14, -18, -32\). This grouping simplifies the process of calculation, as it allows us to perform additions and subtractions in smaller, more manageable steps.

Addition and Subtraction

Addition and subtraction are fundamental operations used to simplify the given expression. By handling one at a time, these operations gradually reduce the expression to a single number.
Steps to simplify using addition and subtraction include:

• Identifying groups of numbers that can be added together, like the positive terms \(14 + 17 + 19 = 50\).
• Similarly, summing up the negative terms while keeping their negative signs, such as \(-12 - 21 - 14 - 18 - 32\) which simplifies in steps to \(-97\).
• Finally, subtract the sum of negative terms from that of the positive terms to find the overall result, making sure you follow the operations correctly to avoid errors.

These basic operations are foundational to simplifying expressions.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which calculations should be performed to ensure consistent results. It's commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In the given exercise, addition and subtraction are the only operations involved, so they should be handled from left to right as they appear. Even though there's no need for parentheses or division, checking the calculation order is critical in verifying the correctness.
By following this order, we avoid common mistakes and ensure the solution is both accurate and efficient. For this problem, consistently applying these rules confirms that the final result of \(-47\) is indeed correct upon re-evaluation of each calculation step.