Problem 52 - Prealgebra

Question

Perform the indicated operations. $$-5(2 x+3)$$

Step-by-Step Solution

Answer

The result of the operation is $$-10x - 15.$$

Step 1: Distribute the Negative Sign

To perform the operation $$-5(2x + 3),$$ we start by distributing the $$-5$$ across each term inside the parentheses. This means you multiply $$-5$$ by each term in the expression inside the parentheses.

Step 2: Multiply $$-5$$ by $$2x$$

Perform the multiplication: $$-5 imes 2x = -10x.$$ So the first term is $$-10x.$$

Step 3: Multiply $$-5$$ by $$3$$

Next, perform the multiplication: $$-5 imes 3 = -15.$$ So the second term is $$-15.$$

Step 4: Combine the Terms

Now, combine the results from the previous steps to write the final expression: $$-10x - 15.$$

Key Concepts

Negative Numbers

Negative numbers are numbers that have a value less than zero. They are often represented with a minus sign (-) in front of them. In mathematics, negative numbers are used in various operations, such as addition, subtraction, and especially multiplication, as we see in the given problem.

When you multiply a negative number by a positive number, the result is always negative. For instance, in the exercise, multiplying \(-5\) by \(2x\) yields \(-10x\). This occurs because negative times positive gives a negative outcome.

Some key points to remember about negative numbers are:

• Multiplying two negative numbers results in a positive number. For example, \(-3 \times -4 = 12\).
• Adding a negative number is equivalent to subtracting its absolute value. For instance, \(-5 + 3\) is the same as \(3 - 5\).
• Negative numbers are essential in representing values below zero, such as temperature or debt.

Multiplication in Algebra

Multiplication in algebra is about finding the total of adding a number to itself a certain number of times. It's crucial in simplifying expressions and solving equations. In the problem \(-5(2x + 3)\), multiplication involves distributing \(-5\) across terms inside the parentheses.

When multiplying numbers and variables, follow these steps:

• Multiply the coefficients (the numbers) together. In \(-5 \times 2\), \(-5\) is the coefficient which multiplies \(2x\) to produce \(-10x\).
• Apply any rules related to signs, remembering that a negative times a positive equals a negative.

This step is essential for simplifying expressions, allowing us to break down problems into manageable parts.

Additionally, multiplication in algebra follows the distributive property, which lets you multiply an outside factor to each term inside the parentheses separately, as seen in the solution where \(-5\) is distributed to both \(2x\) and \(3\).

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions, making them easier to work with. This process involves summing terms that have the same variable raised to the same power. For instance, you can combine \(5x\) and \(-x\) because they both contain the variable \(x\).

In the exercise, after distributing and multiplying, the expression \(-10x - 15\) does not require further combining since the terms are unlike.

• "Like terms" must have identical variable parts. For example, \(3x\) and \(-2x\) are like terms, but \(3x^2\) and \(2x\) are not.
• Constants, or numbers without variables, can be combined on their own, such as \(5 - 3\ = 2\).

The purpose of combining like terms is to streamline polynomial expressions, providing a neater form that makes subsequent calculations more straightforward. It's a fundamental skill in algebra that helps when solving equations and simplifying expressions efficiently.