One Solution, No Solution, Many Solutions

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What does it mean for an equation to have 'No Solution'?

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An equation has 'No Solution' when there are no values for the variable that can satisfy the equation. This often occurs when simplifying the equation leads to a contradiction, such as 0 = 5.

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What does it mean for an equation to have 'One Solution'?

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An equation has 'One Solution' when there is exactly one value for the variable that satisfies the equation. This typically occurs when the variable can be isolated to yield a specific number.

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What does it mean for an equation to have 'Many Solutions'?

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An equation has 'Many Solutions' when any value for the variable satisfies the equation. This usually happens when both sides of the equation are identical after simplification.

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How can you determine if an equation has 'No Solution'?

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To determine if an equation has 'No Solution', simplify the equation and check for contradictions, such as arriving at a statement like 0 = 5.

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How can you determine if an equation has 'One Solution'?

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To determine if an equation has 'One Solution', isolate the variable and solve for it, ensuring that you arrive at a specific numerical value.

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How can you determine if an equation has 'Many Solutions'?

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To determine if an equation has 'Many Solutions', simplify the equation and check if both sides are equivalent, indicating that any value for the variable will satisfy the equation.

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What is an example of an equation with 'No Solution'?

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An example of an equation with 'No Solution' is 18a + 45 = 18a - 37.2, which simplifies to a contradiction.

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What is an example of an equation with 'One Solution'?

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An example of an equation with 'One Solution' is 5z = 25, which simplifies to z = 5.

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What is an example of an equation with 'Many Solutions'?

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An example of an equation with 'Many Solutions' is 2x + 9 = 2x + 9, which is true for all values of x.

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What is the significance of the coefficients in determining the number of solutions?

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The coefficients in an equation can indicate the relationship between the variables. If they are the same on both sides, it may lead to 'Many Solutions'; if they differ, it may lead to 'One Solution' or 'No Solution'.