Find the value of k for which each of the following systems of linear equations has an infinite number of solutions:

$5 x + 2 y = 2 k, 2 (k + 1) x + k y = (3 k + 4) .$

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Solution

$G i v e n space e q u a t i o n s colon 5 x plus 2 y equals 2 k space space space space space space space space space space space space space space space space space space space space space... left parenthesis i right parenthesis 2 left parenthesis k plus 1 right parenthesis x plus k y equals left parenthesis 3 k plus 4 right parenthesis space space... left parenthesis i i right parenthesis H e r e comma space a subscript 1 equals 5 comma space space space space space space space space space space space space b subscript 1 equals 2 comma space c subscript 1 equals 2 k space space space space space space space space space space space a subscript 2 equals 2 left parenthesis k plus 1 right parenthesis comma space b subscript 2 equals k comma space space c subscript 2 equals 3 k plus 4 T h e space g i v e n space p a i r space o f space e q u a t i o n s space h a s space i n f i n i t e space n u m b e r space o f space s o l u t i o n s. rightwards double arrow a subscript 1 over a subscript 1 equals b subscript 1 over b subscript 2 equals c subscript 1 over c subscript 2 rightwards double arrow fraction numerator 5 over denominator 2 left parenthesis k plus 1 right parenthesis end fraction equals 2 over k equals fraction numerator 2 k over denominator 3 k plus 4 end fraction rightwards double arrow fraction numerator 5 over denominator 2 left parenthesis k plus 1 right parenthesis end fraction equals 2 over k rightwards double arrow 5 k equals 2 cross times left parenthesis 2 k plus 1 right parenthesis rightwards double arrow 5 k equals 4 k plus 4 rightwards double arrow k equals 4$
The value k=4 solves the equations hence the value of k is 4.

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