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2 November, 15:00

For which values of k does the system of linear equations have zero, one, or an infinite number of solutions? [Note: not all three possibilities need occur.] (If the answer is an interval of numbers, enter your answer using interval notation. If an answer does not exist, enter DNE.) 8x1 + x2 = 2 kx1 + 9x2 = 18

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  1. 2 November, 15:29
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    For k = 36, there is 0 solutions

    For other values of k, there is 1

    We never got infinite solutions on the system.

    Step-by-step explanation:

    We have 2 unknowns and 2 equations:

    E1 8x1 + x2 = 18

    E2 k*22 x1 + 9 x2 = 18

    If we multiply the E1 by 9 we obtain

    E3 72 x1 + 9x2 = 162

    if we substract E3 with E2 we obtain

    (72 - 2k) x1 = 144

    Thus,

    x1 = 144 / (72-2k)

    That is, if 72-2k = 0, otherwise there is no solution. And 72-2k = 0 when k = 72/2 = 36.

    If k is not 36, then

    x1 = 144 / (72-2k) and we can replace this value to obtain x2 by using E1

    x2 = 18-8x1 = 18 - 8 * (144/72-2k)

    Which is a specific number that depends only on k. Thus,

    for k = 36, there is 0 solutions

    for other values of k, there is unique solution.
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