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Since the top equation creates a circle and the bottom equation creates a line, the possibilities are 0, 1, or 2.

 

To find out how many, solve the equation  3x - y + 1  =  0  for y:  y  =  3x + 1

and substitute this into the first equation:

  x² + y²  =  36     --->     x² + (3x + 1)² =  36

                                x² + 9x² + 6x + 1  =  36

                                     10x² + 6x - 35  =  0

 

If I wanted to find out what the solutions are, I would use the quadratic formula.

Since I only want to know the number of solutions, I will use just the discriminant of the quadratic formula:

   b² - 4·a·c  =  6² - 4·10·-35  =  1436

 

Since the discriminant is positive, there will be two solutions.

If the discriminant were zero, there would be only one solution.

If the discriminant were negative, there would be no solutions.