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:
- Subtract 3 from both sides: \[ 2x = 8 \]
- 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:
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Parentheses: Calculate inside the parentheses first: \[ 3^2 - 1 = 9 - 1 = 8 \] Now the expression becomes: \[ 8 + 2 \times 8 \]
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Multiplication: Next, carry out the multiplication: \[ 2 \times 8 = 16 \] Now the expression is: \[ 8 + 16 \]
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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.
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