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:

x − 3y = 3
3x − 9y = 2

Answer 1

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.