Problem 67 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

26

Step 1: Multiply the First Pair

Multiply the numbers in the first pair, which are \(-6\) and \(-9\). The product of two negative numbers is positive, so: \(-6\) multiplied by \(-9\) is \(54\).

Step 2: Multiply the Second Pair

Multiply the numbers in the second pair, which are \(-7\) and \(4\). The product of a negative and a positive number is negative, so: \(-7\) multiplied by \(4\) is \(-28\).

Step 3: Add the Results Together

Add the results of the products obtained in Step 1 and Step 2. That is, add \(54\) and \(-28\): \(54 + (-28) = 26\).

Key Concepts

Multiplying Integers

When multiplying integers, the sign of the product is determined by the signs of the numbers being multiplied. Here are the basic rules you should remember:

• If you multiply two integers with the same sign, the result is positive.
• If you multiply two integers with different signs, the result is negative.

For example, in the expression \((-6)(-9)\), both numbers are negative. Multiplying these gives a positive result: \(-6 \, \times \, -9 = 54\).
This is because the negative signs cancel each other out.
In another part of the expression, \((-7)(4)\), one number is negative and the other positive, resulting in a negative product: \(-7 \, \times \, 4 = -28\).
Learning these rules will help you tackle all integer multiplication problems with ease!

Adding Integers

Adding integers involves combining both positive and negative values to find a sum. Whether you add positive or negative integers, is important to understand how their signs affect the result.

• When adding two integers with the same sign, add their absolute values and keep the common sign.
• When adding two integers with different signs, subtract the smaller absolute value from the larger one and take the sign of the integer with the larger absolute value.

In the context of expression simplification, after multiplying, we needed to add \(54\) and \(-28\). Here, they have different signs, so we subtract the smaller absolute value from the larger one: \(54 - 28 = 26\).
Since 54 is larger, the result is positive. This method allows us to effectively manage numbers of varying signs.

Negative Numbers

Negative numbers, often found in mathematical problems and real-life scenarios, represent values less than zero. They are crucial in understanding mathematical operations like adding, multiplying, dividing, and subtracting.
In multiplication, knowing the result's sign becomes important:

• Two negative numbers result in a positive product because the negative signs cancel each other.
• One negative and one positive number result in a negative product.

Negative numbers also affect addition and subtraction operations, where determining the resulting sign depends on the size of the numbers involved. This was visible in solving our numerical expression by multiplying and adding negative numbers correctly, resulting in an accurate simplification of the equation.

Order of Operations

The order of operations is a fundamental principle used to determine which calculations to perform first when simplifying expressions. Generally, the acronym PEMDAS helps remember the correct sequence:

• P: Parentheses come first.
• E: Exponents (i.e., powers and roots, etc.)
• M/D: Multiplication and Division from left to right.
• A/S: Addition and Subtraction from left to right.

In our given expression, there are no parentheses or exponents, so we proceed to multiplication first: \((-6)(-9)\) and \((-7)(4)\).
Finally, perform the addition \(54 + (-28)\).
By strictly following this order, we ensure every expression achieves a consistent and correct result.