Given the system:

2x -y = 5...........(1)

x+3y = 12..............(2)

Then, we have a system of two equations and two variables.

We will use the elimination method to solve for x and y.

We will multiply (1) by 3 and add to (2).

==> 6x -3y= 15

==> x + 3y = 12

==> 7x = 27

**==> x = 27/7**

Now to find y, we will substitute into (1).

==> 2x -y = 5

==> 2(27/7) -y = 5

==> 54/7 -5 = y

==> y= (54-35)/7 = 19/7

**==> y= 19/7**

2x-y = 5..(1) and x+3y = 12...(2).

We solve the equations using the analytic geometry.

Here, we a have two linear equations in x and y. We have to solve for the values of x and y. We recast both equations in the slope intercept form, like: y = max+c.

2x-y= 5 => y = 2x-5....................................(1).

x+3y = 12 => y = -(1/3)x + 4......................(2).

We can Easily see that the slopes of the two equations are not same. So they are not parallel. So the equations representing two non parallel lines definitely intesect giving some definite solutions.

At the point of intersection, y coordinates are same. So we equate the right sides of the equations (1) and (2) and solve for x (or coordinate).

2x-5 = -(1/3)x+4.

(2+1/3)x = 5+4.

7x/3 = 9.

(7x/3)*(3/7) = 9*3/7 = 27/7.

x= 27/7.

Therefore, using the first equation, y = 2x-5, where we put x = 27/7, we get: y = 2(27/7) - 5 = 19/7.

Therefore x= 27/7 and y = 19/7.