"*Compute the largest value of x that satisfies x^2 -10x + y^2 - 8y = 8*"

This can be rewritten as (x - 5)^{2} + (y - 4)^{2} = 7^{2}

This is the equation of a circle of radius 7, centred on the point (5,4), so the largest value of x is 5 + 7 = 12.