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Algebra I: Linear Equation-Conditions for Consistency
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The system of equations
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0

(i) is consistent with unique solution, if a1/a2 b1/b2
i.e the lines represented by equations are not parallel.

(ii) consistent with infinitely many solutions, if a1/a2 = b1/b2 = c1/c2
i.e. the lines represented by equations are coincident.

(iii) inconsistent, if a1/a2 = b1/b2 c1/c2
i.e. the lines represented by equations are parallel and non-coincident.
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Example: Determine the values of a and b for which the following system of linear equations infinite solutions.
               4x - (a - 4)y = 2b + 6
               2x -(a - 1)y = 5b - 1
Solution: The condition for infinite solution is a1/a2 = b1/b2 = c1/c2

Therefore, the given system of equations will have infinite number of solutions, if
4/2 = -(a - 4)/-(a - 1) = (2b + 6)/(5b - 1)
2 = (a - 4)/(a - 1) = (2b + 6)/(5b - 1)
2 = (a - 4)/(a - 1) and 2 = (2b + 6)/(5b - 1)
2a - 2 = a - 4 and 10b - 2 = 2b + 6
a = -2 and b = 1.
Hence, the given equations will have infinitely many solutions, if a = -2 and b = 1.
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