1. The problem is to find the length of the diameter of the circle given by the equation $x^2 + y^2 - 6x + 14y = 6$. 2. First, rewrite the equation in standard form by completing the square for both $x$ and $y$ terms. 3. Group $x$ and $y$ terms: $$x^2 - 6x + y^2 + 14y = 6$$ 4. Complete the square for $x$: Take half of $-6$, which is $-3$, and square it: $(-3)^2 = 9$. 5. Complete the square for $y$: Take half of $14$, which is $7$, and square it: $7^2 = 49$. 6. Add these squares to both sides to keep the equation balanced: $$x^2 - 6x + 9 + y^2 + 14y + 49 = 6 + 9 + 49$$ 7. Simplify: $$(x - 3)^2 + (y + 7)^2 = 64$$ 8. This is the standard form of a circle equation with center $(3, -7)$ and radius $r = \sqrt{64} = 8$. 9. The diameter $d$ of a circle is twice the radius: $$d = 2r = 2 \times 8 = 16$$ 10. Therefore, the length of the diameter of the circle is $16$.