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:
x − 3y = 3
3x − 9y = 2
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Solution
The given system of equations may be written as
x - 3y -3 = 0
3x - 9y - 2 = 0
The given system of equations is of the form
`a_1x + b_1y + c_1 = 0`
`a_2x + b_2y + c_2 = 0`
Where `a_1 = 1, b_1 = -3, c_1 = -3`
And `a_2 = 3, b_2 = -9, c_2 = -2`
We have
`a_1/a_2 = 1/3`
`b_1/b_2 = (-3)/(-2) = 3/2`
And `c_1/c_2 = (-3)/(-2) = 3/2`
Clearly, `a_1/a_2 = b_1/b_2 != c_1/c_2`
So, the given system of equation has no solutions.
Concept: Pair of Linear Equations in Two Variables
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