One Solution, No Solution, Many Solutions

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What does it mean if an equation has 'Many Solutions'?

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An equation has 'Many Solutions' if it is true for all values of the variable. For example, the equation Z = Z is true for any value of Z.

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What does it mean if an equation has 'One Solution'?

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An equation has 'One Solution' if there is exactly one value of the variable that makes the equation true. For example, the equation 3x = 9 has one solution: x = 3.

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What does it mean if an equation has 'No Solution'?

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An equation has 'No Solution' if there is no value of the variable that can satisfy the equation. For example, the equation 6x - 3 = 6x + 9 has no solution.

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Example of an equation with 'Many Solutions':

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2x + 9 = 2x + 9.

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Example of an equation with 'One Solution':

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3x = 9.

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Example of an equation with 'No Solution':

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18a + 45 = 18a - 37.2.

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What is the general form of an equation with 'Many Solutions'?

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An equation where both sides are identical, such as a = a.

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What is the general form of an equation with 'One Solution'?

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An equation that can be simplified to a single value, such as x + 5 = 10.

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What is the general form of an equation with 'No Solution'?

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An equation that simplifies to a contradiction, such as 0 = 5.

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How can you identify an equation with 'Many Solutions'?

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If you can manipulate the equation and find that both sides are always equal regardless of the variable's value.