Problem 29 - Elementary Algebra

Question

Translate each sentence to a mathematical statement and then simplify. Find the difference of -16 and -5 .

Step-by-Step Solution

Answer

The difference of -16 and -5 is -11.

Step 1: Understanding the Problem

To find the difference of two numbers, we need to subtract the second number from the first number. In this case, we need to find the difference of -16 and -5.

Step 2: Translate to Mathematical Statement

The word 'difference' implies subtraction. So, the mathematical statement for this exercise is \[-16 - (-5)\].

Step 3: Simplify the Expression

To simplify \[-16 - (-5)\], notice that subtracting a negative is the same as adding its positive, so \[-16 - (-5) = -16 + 5\].

Step 4: Perform the Calculation

Now, calculate the expression: \[-16 + 5 = -11\].

Key Concepts

Mathematical Statements: The Language of Numbers

In everyday language, we often describe mathematical operations with words like "sum," "difference," "product," or "quotient." Translating these words into mathematical symbols is the first step towards solving math problems. A mathematical statement is an expression that uses numbers, operations, and variables to represent a specific scenario or problem.
In our exercise, the term "difference" specifically refers to subtraction. This indicates the operation needed between the numbers. For the problem "Find the difference of -16 and -5," we translate this to the expression:

• "difference" is subtraction,
• so write it as \[-16 - (-5)\]

Understanding how to transform words into a mathematical statement is a key skill that helps bridge everyday language and mathematical problem-solving.

Simplifying Expressions: Clearing Up the Math

Simplifying expressions involves making them as straightforward as possible. It means reducing the complexity of a mathematical statement, while retaining its value. Simplification is a crucial step in solving math problems, as it makes calculations easier to understand and complete.
With the expression \[-16 - (-5)\], there's a special rule: subtracting a negative number is equivalent to adding its positive counterpart. This rule simplifies our expression. So,

• Starting with \[-16 - (-5)\],
• we change it to \[-16 + 5\]

Now, the simplified expression is ready for easy calculation. This step tests your ability to apply mathematical rules and see through complex expressions to find a simpler path to the solution.

Subtraction of Integers: With Negatives in the Mix

Subtraction can become tricky when negative numbers are involved, but understanding the behavior of integers can make it simple. Integers include all positive whole numbers, negative whole numbers, and zero, and recognizing how they interact under subtraction is essential.
When we calculate the difference of \[-16 + 5\], we're looking at how to combine these values:

• Start at -16 on a number line;
• Moving positively by 5 steps gets you to -11.

The result is -11, depicting how subtraction can effectively be visualized as the movement along a number line. Always remember:

• Subtracting a negative, like in \[-16 - (-5)\], turns into addition;
• And positive numbers effectively move you rightward on the number line.

Understanding these basic principles of integer subtraction provides a solid grounding for tackling more complex calculations with confidence.