Problem 29 - Elementary and Intermediate Algebra

Question

Simplify. $$ 5+3 \cdot 7 $$

Step-by-Step Solution

Answer

The simplified result is 26.

Step 1 - Understand Order of Operations

To simplify the expression, it's important to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Step 2 - Identify Operations

Look at the expression \(5 + 3 \times 7\). Here, there is addition (\(5 +\)) and multiplication (\(3 \times 7\)). According to PEMDAS, multiplication should be done before addition.

Step 3 - Perform Multiplication

Calculate the multiplication part of the expression first: \[3 \times 7 = 21\]

Step 4 - Perform Addition

Now, add the result from the multiplication to 5: \[5 + 21 = 26\]

Key Concepts

PEMDAS

When simplifying mathematical expressions, it's crucial to follow the order of operations. This order ensures everyone gets the same result. You can remember this order with the acronym PEMDAS.

• P: Parentheses – Solve anything inside parentheses first.
• E: Exponents – Next, calculate powers or roots.
• M & D: Multiplication and Division – From left to right.
• A & S: Addition and Subtraction – From left to right.

For the expression given:
\(5 + 3 \times 7\),
there are no parentheses or exponents, so we move directly to multiplication and addition.

Simplifying Expressions

Simplifying expressions means breaking them down to their simplest form. This involves following the order of operations and performing any calculations step-by-step.

For example, in the problem \(5 + 3 \times 7\), we start by tackling the multiplication because of the rule we know from PEMDAS.

Simplifying can make dealing with complex equations easier and helps in ensuring accuracy in your calculations. Let's break down our example further.
1. Find the product of \(3\) and \(7\):
\(3 \times 7 = 21\)
2. Add the result to \(5\):
\(5 + 21 = 26\)

Multiplication Before Addition

A key rule in the order of operations is to always perform multiplication before addition. This prevents errors and ensures correct results.

In the expression given, \(5 + 3 \times 7\), multiplication is done before addition, as per the PEMDAS rule.
If we ignored this and simply went from left to right, we’d get a wrong result: \(5 + 3 = 8\) and then, \(8 \times 7 = 56\).

But, by correctly performing multiplication first: We get \(3 \times 7 = 21\) before adding it to \(5\), resulting in \(5 + 21 = 26\).

This example highlights the importance of following the order of operations and why multiplication should be done before addition!