Problem 58 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. What number do you multiply by \(-7\) to get \(-21 ?\)

Step-by-Step Solution

Answer

The number you multiply by \(-7\) to get \(-21\) is 3.

Step 1: Understand the Equation

We are looking to find a number, say \( x \), such that when we multiply it by \(-7\), it results in \(-21\). This can be written as the equation: \(-7x = -21\).

Step 2: Solve for x

To isolate \( x \) in the equation \(-7x = -21\), we need to divide both sides of the equation by \(-7\). This gives us:\[x = \frac{-21}{-7}\]

Step 3: Simplify the Division

When we divide \(-21\) by \(-7\), we simplify it as:\[x = 3\]This is because a negative divided by a negative gives a positive result, and \( 21 \div 7 = 3 \).

Key Concepts

Multiplication Basics

Multiplication is a fundamental arithmetic operation used to calculate the total of one number taken multiple times. It’s like repeated addition but quicker. In the given exercise, we deal with multiplying numbers, specifically finding a number that, when multiplied by -7, results in -21.
Understanding the role of multiplication in solving equations is crucial:

• If you multiply two numbers, the result is called the product.
• When solving a problem like -7x = -21, we are essentially reversing the multiplication to find the original number.
• This is done by dividing the product by the number we originally multiply with, which in this case is -7.

Remember, multiplication can affect how we handle positive and negative numbers due to the rules of signs. In the upcoming sections, you will see how we use this when dealing with negative numbers.

Understanding Negative Numbers

Negative numbers can sometimes be puzzling, but they follow simple rules:

• A negative number is less than zero and opposite a positive number on the number line.
• When you multiply or divide two negative numbers, the result is a positive number.
• If you multiply or divide a negative number with a positive number, the result is always negative.

In solving equations, understanding these properties is crucial. In this exercise, what we're specifically looking at is how dividing -21 by -7 leads to a positive answer. Because -21 and -7 are both negative, their division results in the positive number 3. This aligns with our rule that two negative numbers combined (through multiplication or division) yield a positive product or quotient.

Simplifying Expressions

Simplifying expressions involves using the order of operations to make calculations easier and clear. In our exercise, simplifying means appropriately handling both the multiplication and the negative signs. Here's how you do it:

• Begin with the equation: -7x = -21.
• We aim to isolate the variable x by simplifying the expression using division.
• To "undo" the multiplication by -7, we divide both sides by -7: x = \((-21)\div(-7)\).
• After simplifying, we find: x = 3.

The simplification process assists in making the expression less complex and often helps to reveal the solution more clearly. By dividing and reducing each step properly, the answer becomes straightforward and easy to interpret. Always keep in mind the rules for signs when simplifying as mixing them up can lead to incorrect results.