Question

How many solutions does the following system of equations have `3y-6x=9, 2y-4x=6`?

Answer 1

We have infinite solutions.

Explanation:

The two equations are in fact identical.

If we divide the equation `3y-6x=9` on both sides by same number `3`, we get `y-2x=3`

similarly dividing `2y-4x=6` on both sides by same number `2`, we too get `y-2x=3`

Hence the two equation are identical and actually denote same line `y-2x=3` or `y=2x+3`

Hence, for any `x=x_1`, we can get `y_1=2x_1+3` and `(x_1,y_1)` is a solution to the equation and we can have infinite `(x_1,y_1)` all lying on line `y=2x+3`
graph{2x+3 [-11.54, 8.46, -4.52, 5.48]}
Hence, we have infinite solutions.