Define equivalent expressions.

### Equivalent Expressions

**Definition:**
Equivalent expressions are expressions that represent the same value for all values of the variables involved. For example, \(2(x + 3)\) and \(2x + 6\) are equivalent because they simplify to the same expression regardless of the value of \(x\).

### Methods to Identify Equivalent Expressions

1. **Using Properties of Operations:**
- **Commutative Property:** Changing the order of addition or multiplication does not change the result. For example, \(a + b\) is equivalent to \(b + a\).
- **Associative Property:** Changing the grouping of numbers does not change the result. For example, \((a + b) + c\) is equivalent to \(a + (b + c)\).
- **Distributive Property:** A term can be distributed across an addition or subtraction, i.e., \(a(b + c) = ab + ac\).
- **Combining Like Terms:** Terms with the same variable raised to the same power can be combined. For example, \(3x + 2x = 5x\).

2. **Simplifying Expressions:**
- Simplifying the expressions by performing operations such as addition, subtraction, multiplication, and division can help in identifying equivalence.
- Factoring expressions can also reveal equivalencies. For example, \(x^2 - 1\) factors to \((x - 1)(x + 1)\).

3. **Substituting Values:**
- Evaluating the expressions by substituting specific values for the variables can show whether they yield the same result.
- For example, if \(x = 2\) yields the same result for both expressions, they may be equivalent.

### Evaluating Expressions

To determine if two expressions are equivalent, we can evaluate them:

- Let’s say we have the expressions \(3(x + 2)\) and \(3x + 6\):
1. **Simplifying both:**
- \(3(x + 2) = 3x + 6\) (using the Distributive Property)
2. Therefore, they are equivalent.

- Another example: \(x^2 - 4\) and \((x - 2)(x + 2)\):
1. **Factoring the first expression:** \(x^2 - 4\) factors to \((x - 2)(x + 2)\).
2. Hence, they are equivalent.

By using these methods, you can identify and evaluate expressions to determine if they are equivalent.