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
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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, \(3x + 2x = 5x\).
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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 \((x - 1)(x + 1)\).
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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:
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Let’s say we have the expressions \(3(x + 2)\) and \(3x + 6\):
- Simplifying both:
- \(3(x + 2) = 3x + 6\) (using the Distributive Property)
- Therefore, they are equivalent.
- Simplifying both:
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Another example: \(x^2 - 4\) and \((x - 2)(x + 2)\):
- Factoring the first expression: \(x^2 - 4\) factors to \((x - 2)(x + 2)\).
- Hence, they are equivalent.
By using these methods, you can identify and evaluate expressions to determine if they are equivalent.