The value of k for which the system of equations kx + 10y = k - 5 and 10x + ky = k will have infinite many solutions is:
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Given : system of equations kx + 10y = k - 5 and 10x + ky = k
have infinite many solutions
To Find : The value of k
Solution:
Pair of linear equations
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
Consistent
if a₁/a₂ ≠ b₁/b₂ (unique solution and lines intersects each others)
a₁/a₂ = b₁/b₂ = c₁/c₂ (infinite solutions and line coincide each other )
Inconsistent
if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ ( No solution , lines are parallel to each other)
kx + 10y = k - 5
10x + ky = k
=> k/10 = 10/k = (k-5)/k
k/10 = 10/k
=> k² = 100
=> k = ±10
10/k = (k-5)/k
=> k - 5 = 10
=> k = 15
k/10 = (k-5)/k
=> k² = 10k - 50
=> k² - 10k + 50 = 0
no real solution for k
Hence there is no common solution
so There does not exist any value of k for which infinite many solutions exist
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