Question 320609
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One question per post, please.


If you have a system:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a_1x\ +\ b_1y\ =\ c_1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a_2x\ +\ b_2y\ =\ c_2]


and the coefficients are such that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a_1\ =\ ka_2] and *[tex \LARGE b_1\ =\ kb_2]


but *[tex \LARGE c_1\ \neq\ kc_2]


then you have a system that is inconsistent, i.e., has no solutions. (parallel lines)


If the coefficients are such that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a_1\ =\ ka_2] and *[tex \LARGE b_1\ =\ kb_2]


and *[tex \LARGE c_1\ =\ kc_2]


then you have a system that is consistent and dependent, i.e. there are infinite solutions (the same line)



If the coefficients are such that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a_1\ =\ ka_2] but *[tex \LARGE b_1\ \neq\ kb_2],


then you have a consistent and independent system, i.e. exactly one solution (intersecting lines)


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


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