Determine the value for k for which the system of linear equation has infinitely many solution. {2x-y=2, 4x+ky=4

gemauert79

gemauert79

Answered question

2022-09-18

Determine the value for k for which the system of linear equation has infinitely many solution.
{ 2 x y = 2 4 x + k y = 4

Answer & Explanation

LilsGroolonip86

LilsGroolonip86

Beginner2022-09-19Added 9 answers

2 x y = 2
4 x + k y = 4
Multiply first equation by 2 and subtract it from the second one. So, you have:
2 x y = 2
( k + 2 ) y = 0
If k 2 then y = 0 and x = 1.
If k = 2 then any y is solution (in this case x = ( y + 2 ) / 2
imchasou

imchasou

Beginner2022-09-20Added 2 answers

You have a linear system of two unknowns with two equations.
If these equations are linearly independent you have at most one solution. If these two equations are linearly dependent you have an infinity of solutions.
You can see that by interpreting each equation as a straight line. So the straight lines can be different so they have one or zero intersection point or they are the same and all points on the straight line are solutions.
In this case, to have an infinity of solution you have to find a number λ such as the first eaquation multiplied by λ give the seconde one. So λ 2 = 4, λ ( 1 ) = k and λ 2 = 4 Thus λ 2 = 4 and k = 2

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