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Question
In the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
3x - 5y = 20
6x - 10y = 40
Solution
3x - 5y = 20
6x - 10y = 40
Compare it with
`a_1x + by_1 + c_1 = 0`
`a_1x + by_2 + c_2 = 0`
We get
`a_1 = 3, b_1 = -5, c_1 = -20`
`a_2 = 6, b_2 = -10, c_2 = -40`
`a_1/a_2 = 3/6, b_1/b_2 = (-5)/(-10) , c_1/c_2 = (-20)/(-40)`
Simplifying it we get
`a_1/a_2 = 1/2, b_1/b_2 = 1/2 , c_1/c_2 = 1/2`
Hence
`a_1/a_2 = b_1/b_2 = c_1/c_2`
So both lines are coincident and overlap with each other
So, it will have infinite or many solutions
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