One Solution, No Solution, Infinite Solutions

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What does it mean for a system of equations to have 'No Solution'?

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A system of equations has 'No Solution' when the equations represent parallel lines that never intersect, indicating that there is no set of values that satisfies both equations.

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What does it mean for a system of equations to have 'One Solution'?

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A system of equations has 'One Solution' when the equations represent lines that intersect at exactly one point, indicating a unique solution for the variables.

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What does it mean for a system of equations to have 'Infinite Solutions'?

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A system of equations has 'Infinite Solutions' when the equations represent the same line, meaning any point on the line is a solution.

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How can you determine if a system of equations has 'No Solution'?

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To determine if a system has 'No Solution', check if the equations simplify to a contradiction, such as '0 = 5', indicating parallel lines.

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How can you determine if a system of equations has 'One Solution'?

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A system has 'One Solution' if the equations can be solved to yield a unique pair of values for the variables, typically when the lines intersect at a single point.

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How can you determine if a system of equations has 'Infinite Solutions'?

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A system has 'Infinite Solutions' if the equations simplify to the same equation, indicating that they represent the same line.

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What is an example of a system of equations with 'No Solution'?

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An example is: 2x + 3 = 2x + 5, which simplifies to 0 = 2, a contradiction.

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What is an example of a system of equations with 'One Solution'?

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An example is: x + 2 = 5, which simplifies to x = 3, providing a unique solution.

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What is an example of a system of equations with 'Infinite Solutions'?

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An example is: 3x + 6 = 3(x + 2), which simplifies to the same equation, indicating infinite solutions.

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What role do coefficients play in determining the number of solutions in a system of equations?

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Coefficients determine the slope and position of the lines represented by the equations; different slopes indicate 'One Solution', the same slope with different intercepts indicate 'No Solution', and the same slope and intercept indicate 'Infinite Solutions'.