solve the following system of equations by graphing

3x - 2y = 8
2x - 3y = 7 

Make a table of points for each equationm:

In the first, I arbitrarily chooss -2 for x
and substitute:

   3x - 2y = 8
3(-2) - 2y = 8
   -6 - 2y = 8
       -2y = 14
         y = -7

So one point on the graph of the first equation
in (-2,-7)

I arbitrarily chooss 2 for y
and substitute:

   3x - 2y = 8
 3x - 2(2) = 8
    3x - 4 = 8
        3x = 12
         x = 4

So another point on the graph of the first equation
in (4,2)
 
I arbitrarily chooss 0 for x
and substitute:

   3x - 2y = 8
 3(0) - 2y = 8
    0 - 2y = 8
       -2x = 8
         x = -4

So a third point on the graph of the first equation
in (0,-4).

So plot those points, (-2,-7), (4,2), (0,-4)

and draw a line through them:



In the second equation, I arbitrarily chooss
-1 for x and substitute:

   2x - 3y = 7
2(-1) - 3y = 7
   -2 - 3y = 7
       -3y = 9
         y = -3

So one point on the graph of the second equation
in (-1,-3)

I arbitrarily chooss 1 for y
and substitute:

   2x - 3y = 7
 2x - 3(1) = 7
    2x - 3 = 7
        2x = 10
         x = 5

So another point on the graph of the second equation
in (5,1)
 
I arbitrarily chooss -4 for x
and substitute:

   2x - 3y = 7
2(-4) - 3y = 7
   -8 - 3y = 7
       -3x = 15
         x = -5

So a third point on the graph of the second equation
in (-4,-5).

So plot those points, (-1,-3), (5,1), (-4,-5)
on the same set of axes and draw a line through them:



Through the point where those two lines cross, draw
a vertical line:



Notice that the vertical line crosses the x-axis at
the value 2.  So the x value of the solution is 2.

Also through the point where those two lines cross, draw
a horizontal line:



Notice that the horizontal line crosses the y-axis at
the value -1.  So the y value of the solution is -1.

The solution is therefore (x, y) = (2, -1)

Now let's check to see if this is the correct solution
by substituting in BOTH equations:

Substituting in the first:

     3x - 2y = 8
3(2) - 2(-1) = 8
       6 + 2 = 8
           8 = 8

That checks:

Substituting in the second:

     2x - 3y = 7
2(2) - 3(-1) = 7
       4 + 3 = 7
           7 = 7

That checks, too. So 
we know that this is the
correct solution.

Edwin