Problem 85 - Algebra A Combined Function

Question

Evaluate each expression. \(-7(-2)-3(-1)\)

Step-by-Step Solution

Answer

The expression evaluates to 17.

Step 1: Multiply the first pair of numbers

The first part of the expression is \(-7(-2)\). When we multiply two negatives together, we get a positive result. Thus, \(-7\times(-2)=14\).

Step 2: Multiply the second pair of numbers

The second multiplication involves \(-3(-1)\). Similarly, multiplying two negatives results in a positive: \(-3\times(-1) = 3\).

Step 3: Combine the results

Now, we add the results from the two multiplications: \(14 + 3\). This gives us a final result of \(17\).

Key Concepts

Multiplying Integers

Understanding how to multiply integers is a crucial skill in algebra. When working with integers, the rules of multiplication are straightforward. The main rule to keep in mind is that multiplying two positive numbers yields a positive result; however, multiplying two negative numbers also gives a positive result.

For example, consider multiplying

• Positive imes Positive = Positive (e.g., 3 imes 2 = 6).
• Negative imes Negative = Positive (e.g., ewline -4 imes -5 = 20).
• Positive imes Negative = Negative (e.g., 4 imes -3 = -12).
• Negative imes Positive = Negative (e.g., -2 imes 5 = -10).

The rule for the sign comes first in solving multiplications between integers. Afterward, treat the numbers as absolute values for calculation.

Negative Numbers

Negative numbers can be a bit confusing at first but become easy to manage with practice. They are numbers less than zero, and they appear frequently in mathematical problems and real-life scenarios involving debts, temperatures below zero, etc.

One of the properties of negative numbers is that when you multiply two of them, the result is positive. This happens because of the definition of negative numbers in mathematics, which is often explained through the idea of the opposite or additive inverse. For every negative integer, there is an equivalent positive integer that, when combined, brings the result back to zero.
Some key points about negative numbers include:

• The opposite of a negative number is its positive equivalent (e.g., the opposite of -5 is 5).
• Negative numbers added together will always result in a more negative number (e.g., -3 + -5 = -8).
• When subtracting a negative number, it is similar to adding its positive counterpart (e.g., -7 - (-2) = -7 + 2 = -5).

Understanding these principles helps with solving arithmetic problems and makes working with equations easier.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. They are the foundation of mathematics and are essential for tackling more complex problems.

In the context of evaluating expressions, all integers will typically be subjected to these basic operations, often in a sequence that requires careful order-following. In mathematics, there's a precedence known as "PEMDAS" (Parentheses, Exponents, Multiplication or Division, and Addition or Subtraction) which guides the sequence of these operations.

• Multiplication and division must be treated together, going from left to right as they appear in the expression.
• Addition and subtraction also share equal precedence, and are done after multiplication/division, moving from left to right.

Employing these rules is vital when dealing with expressions involving multiple operations, ensuring that you arrive at the correct answer efficiently, as shown in the exercise where each part is calculated sequentially.