The properties of equality are fundamental rules that justify the manipulation and transformation of equations in mathematics. They ensure that if two expressions are equal, they remain equal when certain operations are performed on both sides of the equation. Here are the main properties of equality:
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Reflexive Property: For any real number \(a\), \(a = a\). This property states that any quantity is equal to itself.
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Symmetric Property: If \(a = b\), then \(b = a\). This property indicates that if one quantity is equal to another, then the second is equal to the first.
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Transitive Property: If \(a = b\) and \(b = c\), then \(a = c\). This property shows that if one quantity is equal to a second, and the second is equal to a third, then the first is equal to the third.
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Addition Property of Equality: If \(a = b\), then \(a + c = b + c\) for any number \(c\). This property allows you to add the same quantity to both sides of an equation without changing the equality.
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Subtraction Property of Equality: If \(a = b\), then \(a - c = b - c\) for any number \(c\). Similar to the addition property, this allows you to subtract the same quantity from both sides of an equation.
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Multiplication Property of Equality: If \(a = b\), then \(a \cdot c = b \cdot c\) for any number \(c\). This property permits you to multiply both sides of an equation by the same nonzero quantity.
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Division Property of Equality: If \(a = b\) and \(c \neq 0\), then \(\frac{a}{c} = \frac{b}{c}\). This property allows you to divide both sides of an equation by the same nonzero quantity.
These properties are essential for solving equations and are widely used in algebra and other areas of mathematics.
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