if 5x-3y=9 and (a-b)x-(a+b-3)y=a-4b represent coincident lines,then values of a and b are
Answers
Given : 5x-3y=9 and (a-b)x-(a+b-3)y=a-4b represent coincident lines
To Find : values of a and b are
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)
5x-3y=9
(a-b)x-(a+b-3)y=a-4b
coincident lines
=> 5/(a - b) = - 3/-(a + b - 3) = 9/(a - 4b)
5/(a - b) = - 3/-(a + b - 3)
=> 5/(a - b) = 3/(a + b - 3)
=> 5a + 5b - 15 = 3a - 3b
=> 2a + 8b = 15 Eq1
5/(a - b) = 9/(a - 4b)
=> 5a - 20b = 9a - 9b
=> 4a + 11b = 0 Eq2
2 * Eq1 - Eq2
=> 5b = 30
=> b = 6
=> a = -33/2
a - b = -33/2 - 6 = -45/2
-(a + b - 3) = - ( -33/2 + 6 - 3) = 27/2
a - 4b = -33/2 - 24 = -81/2
5/(-45/2) = -2/9
-3/(27/2) = -2/9
9/(-81/2) = -2/9
Hence Verified
Values of a and b are -33/2 and 6 respectively.
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