Problem 7 - Algebra 1

Question

Find the product. $$-5 \cdot(-7)$$

Step-by-Step Solution

Answer

The product of -5 and -7 is 35.

Step 1: Identify the numbers and their signs

We have two numbers here, -5 and -7. Both numbers are negative.

Step 2: Apply the multiplication rule

Since both numbers are negative, their product will be positive. So, multiply the absolute values of the numbers. That is \(5*7 = 35\).

Step 3: Apply the sign to the product

As discussed in the previous step, the product of two negative numbers is positive. Therefore, the final answer is +35.

Key Concepts

Negative Numbers

Negative numbers are numbers with a value less than zero, represented with a minus sign in front of the numeral. They are commonly used to denote things like temperatures below zero, debts, or elevations below sea level.
For example, when you see -5, it represents a value that is five units less than zero.

• -5 would be five steps to the left of zero on a number line.
• -7 would be seven steps to the left of zero.

When performing calculations with negative numbers, it is crucial to keep track of their signs as they can change the result of a calculation. For instance, when a negative number is added to a positive number, it is like subtracting a positive value from the negative number's absolute value. Recognizing and understanding negative numbers forms the foundation for more complicated math operations involving these types of numbers.

Multiplication Rules

Understanding the rules of multiplication is key to successfully solving problems that involve multiplying negative numbers. Multiplication has specific rules regarding the signs of the numbers involved.

• If both numbers are positive, the product is positive.
• If both numbers are negative, the product will also be positive. This is because the negatives cancel each other out.
• If one number is negative and the other is positive, the product is negative.

These rules ensure consistency and help in determining the correct sign of the product when two numbers are multiplied together. When multiplying, always determine the sign of the result first before proceeding with the multiplication of absolute values.

Absolute Value

The absolute value of a number is a measure of its magnitude without considering its sign. It is denoted by two vertical lines, like this: \(|n|\). This concept is crucial when dealing with negative numbers in operations like multiplication.

• For instance, the absolute value of -5 is \(|-5| = 5\).
• Similarly, \(|-7| = 7\).

The primary use of absolute values in multiplication is to help isolate the magnitude of numbers to perform the operation without concern for the signs initially. After finding the product of the absolute values, you can then apply the correct multiplication rules to determine the sign of the final result. Understanding absolute values simplifies calculations and clarifies the contributions of each component in an expression.