Problem 57 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. \(-2(5)-(-4)(2)\)

Step-by-Step Solution

Answer

-2

Step 1 - Distribute the Multiplication

First, distribute the multiplication within the parentheses. Calculate ewline \[\begin{equation} -2 \times 5 \text{ and } -(-4) \times 2 ewline -2 \times 5 = -10 ewline -(-4) \times 2 = 4 \times 2 = 8 \end{equation}\]

Step 2 - Combine Results

Next, combine the results from the previous step. Calculate: \[\begin{equation} -10 + 8 = -2 \end{equation}\]

Key Concepts

Multiplication of Integers

Understanding the multiplication of integers is crucial for solving mathematical problems efficiently. Here's a breakdown:

If you multiply two positive integers, the result is positive. For example:
\(3 \times 4 = 12\).

When you multiply a positive integer by a negative integer, the result is negative:
\(3 \times -4 = -12\).

Multiplying two negative integers gives a positive result:
\(-3 \times -4 = 12\).

Remember, the signs of the integers play a significant role:

• Positive × Positive = Positive
• Positive × Negative = Negative
• Negative × Negative = Positive

In the given exercise, we have:
\(-2 \times 5 = -10\)
\(-(-4) \times 2 = 4 \times 2 = 8\).
This explains why the signs of the integers impact the final results.

Distributive Property

The distributive property is a fundamental algebraic principle that simplifies expressions and equations. The property states:
\(a(b + c) = ab + ac\).
Take the problem we have: \(-2(5) - (-4)(2)\).
The steps taken can be broken down as follows:

**Step 1:** Distribute the multiplication within each parenthesis:
\( -2 \times 5 \) and \(-(-4) \times 2\).

This simplification involves basic multiplication first:
\(-2 \times 5 = -10\).
For \(-(-4) \times 2\), remember that \-(-4)\ becomes positive 4, leading to:
\(4 \times 2 = 8\).
Here, the distributive property helps manage and simplify expressions, preparing them for further mathematical operations.

Combining Like Terms

Combining like terms is the process to simplify expressions and equations by summing up terms that are similar. In algebra, like terms are terms that have the same variables raised to the same power. For instance:

• \(3x + 5x\) can be combined to \((3+5)x = 8x\).
• \(2y - y = y\).

In our problem, combining results after distributing the numbers:
We have \(-10 + 8\).
Here, \(-10\) and \(8\) are numbers (or constants), and can be combined directly:
\