Problem 71 - Intermediate Algebra

Question

Simplify each numerical expression. $$ (-6)(-9)+(-7)(4) $$

Step-by-Step Solution

Answer

The result is 26.

Step 1: Multiply the First Pair of Numbers

The expression starts with \((-6) \times (-9)\).When you multiply two negative numbers, the product is positive. This gives:\[(-6) \times (-9) = 54\].

Step 2: Multiply the Second Pair of Numbers

Now, move on to the second pair of numbers \((-7) \times 4\).Since one number is negative and the other is positive, the product will be negative, resulting in:\[(-7) \times 4 = -28\].

Step 3: Add the Results

We now need to add the results of the two multiplications:\[54 + (-28)\].Adding a negative is the same as subtracting the positive number:\[54 - 28 = 26\].

Key Concepts

Multiplication of Negative Numbers

When multiplying two negative numbers, the result is always positive. This is a fundamental principle in mathematics. Think of it as a double negation: two negatives cancel each other out, resulting in a positive. For example, when we multiply

• \((-6) imes (-9) = 54\) -- here, though both numbers are negative, their product is positive.

It helps to visualize this on a number line or through real-life concepts. Imagine walking backward a negative number of steps, which essentially means you move forward.
By understanding this concept, you simplify expressions accurately without second-guessing. This rule applies universally, irrespective of the numbers' magnitude.

Adding Negative Numbers

Adding negative numbers can initially seem confusing, but it's straightforward once you grasp the core idea. When you add a negative number to a positive number, it’s like subtracting its absolute value from the positive number.
For instance, in

• \(54 + (-28) = 54 - 28 = 26\) -- here, adding a negative is akin to subtracting a positive.

This concept can be practically understood by considering financial transactions. If you have \(54 and you owe \)28, your balance is $26. In essence, you deduct debts (negative) from assets (positive).
Recognizing these scenarios helps reinforce the logic behind operations involving negative numbers.

Basic Arithmetic Operations

Basic Arithmetic Operations form the foundation of more complex mathematical reasoning. These operations include addition, subtraction, multiplication, and division.
;

• **Multiplication:** The rules for multiplying numbers change slightly with negative numbers. Similar signs result in a positive product, while dissimilar signs result in a negative product.
• **Addition/Subtraction:** Addition of negative numbers can be seen as subtraction. If the negative number is larger (in absolute terms) than the positive, the result is negative.

Understanding these basics enhances problem-solving skills and facilitates accurate calculation in complex equations. While the operations themselves are simple, their application with negative numbers requires careful consideration of sign rules.