Lets start with the given system of linear equations
In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).
So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.
So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 3 and 9 to some equal number, we could try to get them to the LCM.
Since the LCM of 3 and 9 is 9, we need to multiply both sides of the top equation by 3 and multiply both sides of the bottom equation by -1 like this:
Multiply the top equation (both sides) by 3
Multiply the bottom equation (both sides) by -1
So after multiplying we get this:
Notice how 9 and -9 add to zero, 3 and -3 add to zero, -3 and 3 and to zero (ie ) , and )
So we're left with
which means any x or y value will satisfy the system of equations. So there are an infinite number of solutions