Problem 36 - 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. $$-7+3(6-2)$$

Step-by-Step Solution

Answer

The simplified result of the expression is 5.

Step 1: Evaluate Inside the Parentheses

Begin by simplifying any expressions inside parentheses. Here, the expression inside the parentheses is \(6 - 2\). Calculate \(6 - 2\), which equals 4. So, the expression becomes \(-7 + 3(4)\).

Step 2: Perform Multiplication

Proceed by performing the multiplication operation, since multiplication comes before addition and subtraction in the order of operations. Multiply \(3\) by \(4\), which equals \(12\). Now the expression is simplified to \(-7 + 12\).

Step 3: Perform Addition

Finally, add \(-7\) to \(12\). This is a straightforward addition operation with negative numbers. \(-7 + 12 = 5\).

Key Concepts

Understanding Parentheses in Math

In mathematics, using parentheses is a way of prioritizing operations. When you see parentheses, it's a signal to solve the expressions within them before addressing anything outside. This is because operations inside parentheses take precedence over all other operations.
Consider this as a "first-come, first-served" principle for the numbers and operations inside. As an example, in the expression \(-7 + 3(6-2)\), the operation inside the parentheses is \(6-2\). This is why the first step is to simplify by calculating \(6 - 2\) which equals 4.
By dealing with the parentheses first, you ensure that you are following the rules that maintain mathematical clarity and accuracy. Simplifying expressions step by step allows you to systematically tackle each part of a mathematical puzzle without other operations interfering prematurely.

The Role of Multiplication in Order of Operations

After dealing with parentheses, the next priority in the order of operations is multiplication. This crucial step should be handled before moving on to addition or subtraction. Imagine multiplication as a secondary key in unlocking the solution after handling what’s inside the parentheses.
In our example expression \(-7 + 3(4)\), once \(6 - 2\) is simplified to 4, the multiplication comes into play. You then multiply \(3\) by \(4\), resulting in 12. This step is essential because it transforms parts of the equation into values you can easily handle afterward.
By performing multiplication at this stage, you've simplified the expression as much as possible before dealing with the remaining operations. This step-by-step reduction is what makes complex expressions manageable.

Solving Addition and Subtraction Last

Finally, after tackling operations inside parentheses and performing any necessary multiplications, we come to addition and subtraction. These two operations are of equal priority and should be solved from left to right in the order they appear.
In the expression \(-7 + 12\), you can now perform the addition. Here's where handling negative and positive numbers becomes significant. Calculating \(-7 + 12\) involves combining a negative and a positive number, resulting in 5.
Addition and subtraction are naturally simpler once earlier operations are completed, but it’s important to keep the order straight. Understanding this sequence not only aids in solving equations correctly but helps build a strong foundation for tackling more complex arithmetic in the future.