Problem 40 - 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. $$6(-1)+2(-7)$$

Step-by-Step Solution

Answer

The expression simplifies to -20.

Step 1: Apply Multiplication First

According to the order of operations, multiplication should be performed before addition and subtraction. Identify the multiplication operations in the expression and solve them. The expression is \(6(-1) + 2(-7)\). Begin by calculating \(6(-1)\) and \(2(-7)\). \[ 6(-1) = -6 \]\[ 2(-7) = -14 \]

Step 2: Substitute and Simplify

Now substitute the results of the multiplication back into the expression. The expression becomes:\(-6 - 14\).

Step 3: Perform Addition/Subtraction

With the multiplication completed, move on to continue the operations as indicated from left to right. Simplify the expression by adding \(-6\) and \(-14\). \[ -6 - 14 = -20 \]

Key Concepts

The Multiplication Rule

The multiplication rule is an important part of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). When evaluating expressions, always perform multiplication before moving on to addition or subtraction. This ensures you follow the correct mathematical rules and get accurate results. Imagine you're preparing a cake: it's crucial to measure and mix your ingredients correctly before you bake. Similarly, in math, multiplication must be completed before stirring in any other operations. In our example, the expression is \(6(-1) + 2(-7)\). Start by handling the multiplication parts:

• \(6(-1)\) translates to \(-6\)
• \(2(-7)\) translates to \(-14\)

By conducting these steps first, you create a solid foundation for the rest of the calculations.

The Addition Rule

After solving any multiplication in an expression, utilize the addition rule. According to the order of operations, after multiplication, you address addition and subtraction as they appear from left to right. It's crucial to follow this directional cue to avoid mistakes and ensure proper simplification. In our given exercise, after multiplying, the expression became \(-6 + (-14)\). Typically, when adding numbers, if both are negative, you simply add their absolute values for a negative result:

• \(-6 + (-14) = -20\)

Follow the order, just like singing the song in the correct order to remember the lyrics accurately.

The Subtraction Rule

Subtraction is the next in line following multiplication and addition, yet it often works hand-in-hand with addition, especially in expressions containing negative numbers. The rule applies that subtraction follows the same left-to-right processing as addition does. In the realm of negative and positive numbers, subtraction can sometimes mean adding a negative number. In the example \(-6 - 14\), this can be considered as \(-6 + (-14)\), reinforcing the role of addition with negative numbers. When subtracting, convert the operation into adding a negative:

• Start at -6 and move 14 steps more negative, resulting in \(-20\).

This straightforward technique helps simplify the confusion often associated with handling both addition and subtraction.