Question 310664
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Two parallel lines, no solution.


How did I know?


If you have:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \alpha_1x_1\ +\ \beta_1x_2\ =\ \gamma_1]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \alpha_2x_1\ +\ \beta_2x_2\ =\ \gamma_2]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \alpha_1\ =\ k\alpha_2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \beta_1\ =\ k\beta_2]


and 


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \gamma_1\ \neq\ k\gamma_2]


where


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ k\ \in\ \mathbb{Z}]


Then you have two parallel lines and no solution.


On the other hand, if you have all of the above except that


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \gamma_1\ =\ k\gamma_2]


then you have two representations of the same line and therefore an infinite number of solutions.



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John
*[tex \LARGE e^{i\pi} + 1 = 0]
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