Here’s one way to apply Newton’s Method.
g(x,y)=0⇒y=(x²+24x-34)/9,
y-2=(x²+24x-34)/9-2=(x²+24x-52)/9=(x-2)(x+26)/9.
Substitute for y in f(x,y)=0:
(x-2)²+2(x-2)²(x+26)²/81-100=0,
(x-2)²+2(x-2)²(x+26)²/81-100=0,
(x-2)²(1+2(x+26)²/81)-100=0,
(x-2)²(2x²+104x+1433)/81-100=0.
Differentiate:
2(x-2)(2x²+104x+1433)/81+(x-2)²(4x+104)/81.
x₀=4,
x₁=x₀-[(x₀-2)²(2x₀²+104x₀+1433)/81-100]/[2(x₀-2)(2x₀²+104x₀+1433)/81+(x₀-2)²(4x₀+104)/81]
More to follow...