Problem 59 - 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. Subtract \(-3\) from the product of \(-5\) and 4

Step-by-Step Solution

Answer

The simplified result is -17.

Step 1: Identify the Scenario

We start by identifying the mathematical operation in the given expression: 'Subtract -3 from the product of -5 and 4'. This involves both multiplication and subtraction operations.

Step 2: Calculate the Product

First, find the product of -5 and 4. To do this, multiply -5 by 4, which yields -20 because a negative number multiplied by a positive number results in a negative number.

Step 3: Perform the Subtraction

The expression requires us to subtract -3 from the product we found. Therefore, compute -20 - (-3). Subtracting a negative number is the same as adding its positive, so this simplifies to -20 + 3.

Step 4: Calculate the Final Result

Add the numbers together: -20 + 3 equals -17. This gives us the final simplified result for the expression.

Key Concepts

Understanding Multiplication with Negative Numbers

When multiplying numbers, it's important to consider the signs of the numbers involved. The rule is simple:

• Multiplying two positive numbers results in a positive product.
• Multiplying two negative numbers also gives a positive product.
• However, multiplying a positive number by a negative number yields a negative product.

This rule is crucial since it directly affects the sign of the product. In our exercise, we multiplied \(-5\) by \(4\). Here, \(-5\) is negative and \(4\) is positive, so the product is \(-20\), which is negative.
Understanding how to handle these signs when multiplying is key to getting correct results in many mathematical problems.

The Role of Subtraction

Subtraction is a staple in mathematics, often introduced early but it has its nuances, especially with negative numbers.

• When subtracting a positive number, you move left on the number line, making the result smaller.
• Subtracting a negative number is effectively adding a positive. This is because two negatives negate each other.

In our step-by-step solution, we needed to subtract \(-3\) from \(-20\). Mathematically, \(-20 - (-3)\) becomes \(-20 + 3\).
This concept can sometimes be confusing, but remembering that subtracting a negative is like adding a positive makes it easier.
This step highlights the importance of understanding the sign rules of subtraction, especially when dealing with negative numbers.

Simplification Through Order of Operations

Simplification makes expressions easier to understand and solve, riding on the back of the order of operations. This is especially crucial when multiple operations are involved, as in our exercise.The order of operations is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Each operation should be carried out in this order
In the given exercise:

• We first addressed multiplication, computing the product of \(-5\) and \(4\) to get \(-20\).
• Next, we simplified the subtraction operation by understanding \(-20 - (-3)\) as \(-20 + 3\).

Following this logical sequence ensured the expression was simplified correctly to \(-17\). Every step adhered to the order of operations, leading to the accurate solution. Understanding and applying these rules correctly are the backbone of efficiently solving mathematical problems.