There are two solutions, as you can see below. The two black lines are
3x + 4y = 12 and 4x +3y = 9.  The blue line is 3x + y = 7, and you see
that both circles have their centers on the blue line, and they touch the
two black lines.   I will find the equation of the bigger circle.  You
 find the equation of the smaller circle.  It is found the same way.




We draw the two lines (in black) that the circles must touch (be tangent to):



In order that a circle be tangent to two lines, its center must lie on one of
the bisectors of the angles between the two lines.  Since the perpendicular
distance from every point on the angle bisector to both lines must be equal,
We equate their normal forms and simplify to find the equation of the two angle
bisectors



Since their denominators are the same, 



Using the +,





   <--- one angle bisector

Using the -,











   <--- the other angle bisector

Next we graph the two angle bisectors, one in green, and one in red:



The centers of the circles must be on these bisectors.

We are also told that the circles must have its center on the line
3x+y=7

We draw that line (in blue):



Next we find the point where the green and blue lines intersect,
by solving this system of equations:



Their solution is (1,4), so that is the center of one of the circles.
Next we find the radius of the circle with that radius.  We do that
by finding the perpendicular distance from the center (1,4) to either one
of the original two black lines


That works out to be 7/5, so the radius is 7/5 and the center is (1,4)

Therefore one answer is found by substituting in the standard form of
a circle:



  <--that's one of the answers.



You can find the other solution the same way by finding the center where the
blue line intersects the red line. Here is the graph of both circles



Edwin