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

Step-by-Step Solution

Answer

The simplified expression is -47.

Step 1: Group Positive and Negative Numbers

First, let's group all the positive numbers together and all the negative numbers together. The positive numbers in the expression are 14, 17, and 19. The negative numbers are -12, -21, -14, -18, and -32.

Step 2: Calculate the Sum of Positive Numbers

Now we'll add up the positive numbers: \[ 14 + 17 + 19 = 50 \]

Step 3: Calculate the Sum of Negative Numbers

Next, we add the negative numbers: \[ (-12) + (-21) + (-14) + (-18) + (-32) = -97 \]

Step 4: Combine Results

Now we need to find the total sum by combining the results from the positive and negative sums:\[ 50 + (-97) = 50 - 97 = -47 \]

Key Concepts

Properties of Addition

When working with numerical expressions, understanding the properties of addition can simplify calculations. The main properties are:

• Commutative Property: This states that the order of numbers does not matter when adding. For example, in the sum of numbers like 14 and 19, both 14 + 19 and 19 + 14 give the same result.
• Associative Property: This property allows us to group numbers differently without changing the sum. Thus, (14 + 17) + 19 is the same as 14 + (17 + 19).
• Identity Property: Adding 0 to any number does not change its value, so it serves as an identity element in addition.

By leveraging these properties, calculations can be streamlined and often made without writing them out sequentially. For complex problems, grouping numbers systematically using these properties reduces errors and speeds up the process.

Grouping Numbers

Grouping numbers is a fundamental technique in simplifying expressions. This approach involves identifying parts of an expression that can work together conveniently. In the problem given, we separate positive numbers and negative numbers before proceeding with calculations.

By grouping positive numbers (e.g., 14, 17, 19) and calculating their sum first, you simplify the handling of values without signs initially causing confusion. Similarly, we group all negative elements (-12, -21, -14, -18, -32).

This helps in managing the expression in smaller steps. The smaller parts are easier to understand and compute. This step-by-step handling is logical and significantly reduces complexity, making it easier to perform arithmetic operations accurately.

Positive and Negative Numbers

Understanding positive and negative numbers ensures accurate calculations. Positive numbers are numbers greater than zero and often appear without a sign or with a plus (+) sign. Negative numbers are less than zero and show a minus (-) sign.

When adding negative numbers, think of it as moving left on a number line. The opposite goes for positive numbers. Say, adding -10 to -5 moves further left into negative territory.

Subtracting a negative number is trickier. It becomes an addition: subtracting -5 means adding +5. Meanwhile, adding a positive number increases value, and adding a negative number decreases it. These rules guide you through effectively managing mixed expressions of positives and negatives.

Sum of Integers

The sum of integers involves adding a mixture of negative and positive numbers. Strategies to compute these easily include:

• Adding Positive Integers: Simply sum all positive numbers as usual (e.g., 14 + 17 + 19 = 50).
• Adding Negative Integers: Treat them like debts. Compute their sum as their positive equivalent would be added, but keep it negative (e.g., -12 + -21 + -14 + -18 + -32 = -97).
• Combining Sums: After finding the total for positives (50) and negatives (-97), simply sum these results as final steps in calculations: 50 + (-97) = -47.

Approaching integers this way allows clear steps from separating types of numbers to combining their sums confidently. This simplification is key to mastering integer operations in algebraic expressions.