Well, if it touches both axes we know that there is a solution when
x=0 and when
y=0.
This is much more easily visualised if you draw the line
y=3x−4 and you essentially are looking for a point on the line where you can draw a circle and have it just touch both axes - this point will be unique. However, for the equation
y=x you should be able to see there are an infinite number of solutions. For example, if the centre was
(1,1) and you had the radius as
1 it would satisfy the problem, and in fact any centre of
(p,p) with radius
p works - there are infinitely many solutions. However, because this line is not going through the origin we're looking for a point on the line
y=3x−4 which is at equal distances from the y-axis and the x-axis. Again, if you don't understand this, you need to draw it. Clearly, the centre
(5,11) is on the line
y=3x−4 - but can we form a circle which touches both the x-axis and y-axis?