Problem 24 - Algebra for College Students

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$16-14-13-18+19+14-17+21$$

Step-by-Step Solution

Answer

The simplified expression equals 8.

Step 1: Identify Like Terms

Group the positive numbers and the negative numbers separately to make calculations simpler. This will help you take advantage of the associative and commutative properties. Positive numbers: 16, 19, 14, 21 Negative numbers: -14, -13, -18, -17.

Step 2: Sum the Positive Numbers

Add up all the positive numbers. First, add 16 and 14: \[ 16 + 14 = 30 \]Then add 19 and 21: \[ 19 + 21 = 40 \]Now add these results: \[ 30 + 40 = 70 \].

Step 3: Sum the Negative Numbers

Add up all the negative numbers.First, add -14 and -13: \[ -14 - 13 = -27 \]Then add -18 and -17: \[ -18 - 17 = -35 \]Now add these results:\[ -27 - 35 = -62 \].

Step 4: Combine the Results

Add the sums of the positive and negative numbers to get the final result.Combine the results from Steps 2 and 3:\[ 70 + (-62) = 70 - 62 = 8 \].

Key Concepts

Associative Property

The associative property is a fundamental concept in mathematics that helps simplify complex expressions. This property applies to both addition and multiplication. It states that the grouping of numbers does not affect their sum or product. In simpler terms, when adding or multiplying numbers, how you group them in brackets does not change the result.

For example, in addition, the associative property is expressed as:

• \((a + b) + c = a + (b + c)\)

This means that if you are adding three or more numbers, you can choose where to place the parentheses without changing the outcome. This property is particularly useful when trying to make calculations easier.

In our exercise, when adding the positive numbers: first the numbers 16 and 14 are grouped and added, then 19 and 21. After that, the sums are combined, showcasing the associative property in action.

Commutative Property

The commutative property, much like the associative property, applies to both addition and multiplication. This property states that the order in which numbers are added or multiplied does not affect the final result. In simple terms, swapping the positions of numbers in these operations will yield the same outcome.

The commutative property of addition can be expressed as:

• \(a + b = b + a\)

In multiplication, it is also true: \(a \times b = b \times a\). This property is particularly helpful when rearranging the terms of an expression to simplify calculations.

• For instance, when adding 14 to 16 or 16 to 14, the sum remains 30.

By strategically reordering terms, we can often make the calculation process more intuitive.

Addition of Integers

Adding integers can sometimes seem challenging, especially when dealing with both positive and negative numbers. It is important to understand how to correctly add these numbers to simplify expressions.

Here’s a simple guide:

• Adding two positive integers: Simply add the absolute values.
• Adding two negative integers: Add the absolute values and assign a negative sign to the result.
• Adding a positive and a negative integer: Find the difference between their absolute values and keep the sign of the larger absolute value.

In our exercise, adding positive numbers, such as 16 + 14 to get 30, and negative numbers, like -14 and -13 to get -27, demonstrates how these rules apply. The results of these additions are then combined, taking the sign and values into account.

Grouping Terms

Grouping terms is an essential strategy used to streamline arithmetic operations and simplify expressions. The goal is to arrange terms so that calculations can be performed more easily. By strategically grouping numbers, often through the use of parentheses, we can utilize the associative and commutative properties more effectively.

In the provided problem, grouping involves separating positive and negative numbers. This allows us to deal with sums and differences more efficiently:

• Positive Numbers: Group numbers like 16, 19, 14, and 21 together to perform addition.
• Negative Numbers: Similarly, gather numbers such as -14, -13, -18, and -17 for subtraction.

By first handling these groups separately, then combining their results, you simplify the overall expression, making it easier to reach a solution. Grouping terms is a vital step in managing complex calculations in a step-by-step manner.