Question

Equation Define and illustrate

Same thing with Order of Operations

Answer 1

Sure! Let's break down the concepts of "equation" and "order of operations."

### Equation

**Definition:**
An equation is a mathematical statement that asserts the equality of two expressions. It typically contains variables (letters representing numbers), constants, and operators (such as +, −, ×, and ÷). An equation is formed when an expression on the left side is equal to an expression on the right side, often represented in the form:

\[
\text{Expression}_1 = \text{Expression}_2
\]

**Illustration:**
For example, consider the equation:
\[
2x + 3 = 11
\]
In this equation, \(2x + 3\) and \(11\) are two expressions. The goal is to solve for the variable \(x\).

**Steps to solve:**
1. Subtract 3 from both sides:
\[
2x = 8
\]
2. Divide both sides by 2:
\[
x = 4
\]

So, the solution to the equation is \(x = 4\).

### Order of Operations

**Definition:**
The order of operations is a set of rules that dictates the correct sequence to evaluate a mathematical expression involving multiple operations. The standard order can be remembered using the acronym PEMDAS:

- **P** - Parentheses first
- **E** - Exponents (i.e., powers and square roots, etc.)
- **M** - Multiplication and Division (from left to right)
- **A** - Addition and Subtraction (from left to right)

**Illustration:**
Consider the expression:
\[
8 + 2 \times (3^2 - 1)
\]

Using the order of operations, we would solve it as follows:

1. **Parentheses:** Calculate inside the parentheses first:
\[
3^2 - 1 = 9 - 1 = 8
\]
Now the expression becomes:
\[
8 + 2 \times 8
\]

2. **Multiplication:** Next, carry out the multiplication:
\[
2 \times 8 = 16
\]
Now the expression is:
\[
8 + 16
\]

3. **Addition:** Finally, perform the addition:
\[
8 + 16 = 24
\]

So, the value of the expression \(8 + 2 \times (3^2 - 1)\) is **24**.

### Summary
- An **equation** expresses equality between two mathematical expressions and can involve solving for unknown variables.
- The **order of operations** provides a structured method for evaluating expressions correctly, ensuring that calculations are consistent and accurate.