Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$14-2 \cdot 6+3$$

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Carry out Multiplication

First, deal with the multiplication involved in the equation. This is the \(2 \cdot 6\) term. This comes out as 12. The revised equation becomes \(14 - 12 + 3\).

Step 2: Subtraction and Addition (from left to right)

Next, perform the subtraction first: \(14 - 12 = 2\). Now the equation becomes \(2 + 3\). Then, carry out the addition: \(2 + 3 = 5\).

Key Concepts

Multiplication

When you're working on simplifying mathematical expressions, it's crucial to understand the importance of multiplication and its position in the order of operations. In any arithmetic expression, multiplication takes precedence over addition and subtraction. For instance, in the original exercise involving the expression \(14 - 2 \cdot 6 + 3\), the multiplication \(2 \cdot 6\) is tackled first.

Let's break it down further with important points:

• Spot the multiplication step in any expression. In this case, it's \(2 \cdot 6\).
• Solve this part of the equation first. Here, \(2 \cdot 6\) results in 12.
• Once you have solved the multiplication, rewrite the expression incorporating the solution. Changing \(14 - 2 \cdot 6 + 3\) to \(14 - 12 + 3\) after resolving the multiplication.

Understanding and applying multiplication first is a cornerstone in simplifying expressions correctly.

Addition and Subtraction

After dealing with multiplication, it's time to address addition and subtraction. These operations are performed from left to right, without any inherent preference between the two.

Let's explore step by step:

• Identify the sequence of operations after any multiplication is complete. In our example, the expression becomes \(14 - 12 + 3\).
• From left to right, first handle subtraction unless the problem designates another rule. Here, compute \(14 - 12\) to get 2.
• Next, you continue with the addition, solving \(2 + 3\), which gives us the final value of 5.

By following these pathways, you maintain the integrity of mathematical operations, leading to accurate results.

Simplification

The process of simplification in mathematics is about reducing an expression to its most basic form without altering its value. In dealing with the expression provided, simplification is applied through the correct sequence of operations.

Here's the approach:

• First, simplify using multiplication as explained earlier, turning the expression into \(14 - 12 + 3\).
• Then, address the combined actions of subtraction and addition from left to right, obtaining \(2 + 3 = 5\).
• The result, 5, is the simplest form of the original expression.

By systematically following multiplication first, then addition and subtraction, we can simplify expressions effectively, which is an essential mathematical skill for solving complex problems.