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 set of values that satisfies both equations.

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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 every point on the line is a solution to the system.

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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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To determine if a system has 'One Solution', check if the equations simplify to a true statement with different slopes, indicating intersecting lines.

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

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To determine if a system has 'Infinite Solutions', check if the equations simplify to the same equation, indicating that they represent the same line.

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What is the significance of the slopes of lines in determining the number of solutions?

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The slopes of the lines determine their relationship: if slopes are equal and y-intercepts are different, there is 'No Solution'; if slopes are different, there is 'One Solution'; if slopes and y-intercepts are equal, there are 'Infinite Solutions'.

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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. This simplifies to 0 = 2, which is 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 = 3 and x = 1. These equations intersect at the point (1, 3).

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

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An example is: 2x + 4 = 2(x + 2). Both equations simplify to the same line.