SOLUTION: Find the equation of the circle that inscribed in the triangle whose sides are 3x - 4y = 19, 4x + 3y = 17 and x + 7 = 0.

Hello,

I don't know what is your level in Math (what is your grade and the level of knowledge).


What I do know that this problem is hard, is of the high level and requires high level technique (virtuoso technique) 
and high level (adequate) understanding.


So, I will ASSUME that your educational level corresponds to the level of the problem (otherwise why did you start it ?)


Therefore, I will give you only the idea of solution, but will not go into details.


So, first.  Notice that the first and the second lines are perpendicular.

            Also notice that the third line is vertical.

            So, you have right-angled triangle.



Second, you need to find the center of the inscribed circle. 

            The center lies at the intersection of the angle bisectors.

            As the first angle bisector, it is naturally to take the angle bisector of the right angle.

            So, find the intersection point of the first two lines (solve the system of 2 equations in two unknowns).

            Next, you know the slopes of these two lines (from their equations).
            They are tangents of some canonical angles on the coordinate plane.
            By knowing tangents of those angles, find the tangent of the angle bisector of the straight angle. 
            It will give you the slope of that angle bisector.

            In this way you can obtain the equation of the bisector of the right angle.
            As you CAN do it in principle, I will assume that you can do it in reality.



Now, third. You can do the same (or the similar) for the angle bisector of any other acute angle of the triangle.

            In this way you can obtain the equation of the bisector of the second angle.
            As you CAN do it in principle, I will assume that you can do it in reality.



Step #4. Having equations of the angle bisectors, you can find their intersection.
             It will be the center of the circle.


             By having coordinates of the circle, you can easily complete the solution.

     What I described above, is not the unique way to solve the problem.
     There is another way.
     It is to calculate the measures of all three sides;
           then to use the property of the angle bisector in any triangle:
               
                it divides the opposite side proportionally to adjacent sides.
          
           Using this property, you can find the intersection point for every angle bisector at the opposite side.
           Having it, you can construct the equations for each angle bisector.