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Question

For what value of k does the system of equations x+2y=3,5x+ky+7=0 have(i) a unique solution,(ii) no solution?

Also,show that there is no value of k for which the given system of equations has infinitely many solutions.

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Solution

The given equation be as:

x + 2y -3 = 0....(1)

5x + ky + 7 = 0....(2)

We have,
a subscript 1 equals 1 space space semicolon space space b subscript 1 equals 2 space space semicolon space space c subscript 1 equals negative 3 space space semicolon a subscript 2 equals 5 space space semicolon space space b subscript 2 equals k space space semicolon space space c subscript 2 equals 7 space space space space space space semicolon

(i) For unique solution;

a subscript 1 over a subscript 2 not equal to b subscript 1 over b subscript 2
1 fifth not equal to 2 over k space space semicolon k not equal to 10

(ii)For No solution;
a subscript 1 over a subscript 2 equals b subscript 1 over b subscript 2 not equal to c subscript 1 over c subscript 2 space space semicolon 1 fifth equals 2 over k not equal to fraction numerator negative 3 over denominator 7 end fraction space space space space semicolon w h i c h space g i v e s semicolon k equals 10 space

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