Question 56846
Find the coordinates of the vertices of the triangle whose sides are contained in the lines whose equations are 5x - 3y = -7, x + 2y = 9, and 3x - 7y = 1.

5x - 3y = -7
 x + 2y = 9 --> x = -2y + 9
3x - 7y = 1

5(-2y + 9) - 3y = -7
-10y + 45 - 3y = -7
-13y + 45 = -7
     - 45 on both sides
-13y = -52
    divide by -13 on both sides
y = 4

put 4 back into the equation to find x:
5x - 3(4) = -7
5x - 12 = -7
    add 12 to both sides:
5x = 5
x = 1

(1, 4)

check answers:
5x - 3y = -7 --> 5(1) - 3(4) = 5 - 12 = -7  (correct)
 x + 2y = 9  --> 1 + 2(4) = 1 + 8 = 9  (correct)
:
You were right about the first vertex, but there are three vertices in a triangle.  
Look at the graph:
{{{graph(300,200,-10,10,-10,10,(-5x-7)/-3,(-x+9)/2,(-3x+1)/-7)}}}
The red line is: 5x-3y=-7
The green line is: x+2y=9
The blue line is: 3x-7y=1
You found 1, the intersection of:
5x-3y=-7 and x+2y=9
Now you have to find the intersection of:
5x-3y=-7 and 3x-7y=1
Since you know how to do that, I'll just give you and answer to check yourself against.  If you don't get that answer, let me know and I'll elaberate: (-2,-1)
Finally you have to find the intersection of:
x+2y=9 and 3x-7y=1  Check your answer against: (5,2).
Let me know, if you need more information.  I think you are on the right track and just needed a nudge.
Happy Calculating!!!