To solve this system of equations, we can start by solving for one variable in terms of the others.

From the first equation, we have x/3 = y/7. Cross multiplying, we get 7x = 3y.

From the second equation, we have y/7 = z/5. Cross multiplying, we get 5y = 7z.

Now we can substitute these expressions into the third equation.

x^2 - y^2 + z^2 = -60
(x/3)^2 - (7x/3)^2 + (5y/7)^2 = -60
(x^2/9) - (49x^2/9) + (25y^2/49) = -60
(1/9 - 49/9 + 25/49)(x^2) + (25/49)(y^2) = -60
(-24/49)(x^2) + (25/49)(y^2) = -60

Now we can substitute the expression for y in terms of x from the first equation.

(-24/49)(x^2) + (25/49)((7x/3)^2) = -60
(-24/49)(x^2) + (25/49)((49x^2)/9) = -60
(-24/49)(x^2) + (25/49)(49x^2)/9 = -60
(-24/49)(x^2) + (25/9)(x^2) = -60
[(25/9) - (24/49)](x^2) = -60
[(1225/441) - (216/441)](x^2) = -60
(1009/441)(x^2) = -60

Now we can solve for x^2.

x^2 = (-60)(441/1009)
x^2 = -2646/1009

Similarly, we can solve for y^2 and z^2 using the expressions we derived earlier.

y^2 = (49/25)(7z/5)^2
y^2 = (49/25)(49z^2/25)
y^2 = 2401z^2/625

Substituting these expressions into the third equation:

(-2646/1009) - (2401z^2/625) + z^2 = -60
(-2646/1009) - (2401z^2/625) + (625z^2/625) = -60
(-2646/1009) + (3856z^2/625) = -60
(625*(-2646) + 3856z^2)/(1009*625) = -60
-1653750 + 3856z^2 = -60(1009*625)
-1653750 + 3856z^2 = -60540000
3856z^2 = -60540000 + 1653750
3856z^2 = -58886250
z^2 = -58886250/3856

Since the square of a real number cannot be negative, there is no real solution to this system of equations.