[1]#color(white)("XXXX")##(x-3)^2+(y-5)^2=25#

[2]#color(white)("XXXX")##x^2+(y-1)^2=100#

Expanding the square for [1]

[3]#color(white)("XXXX")##x^2-6x+9 + y^2-10y+25 = 25#

Simplifying

[4]#color(white)("XXXX")##x^2-6x+y^2-10y = -9#

Expanding the square for [2]

[5]#color(white)("XXXX")##x^2 +y^2-2y+1 = 100#

Simplifying

[6]#color(white)("XXXX")##x^2+y^2-2y= 99#

Subtract [4] from [6]

[7]#color(white)("XXXX")##6x +8y = 108#

Simplify

[8]#color(white)("XXXX")##y = (54-3x)/4#

Substituting #((54-3x)/4)# for #y# in [2]

[9]#color(white)("XXXX")##x^2+((54-3x)/4 -1)^2 = 100#

or

[10]#color(white)("XXXX")##x^2+(50/4-(3x)/4)^2 = 100#

Expanding the square

[11]#color(white)("XXXX")##x^2 + (2500/16-(300x)/16+(9x^2)/16) = 100#

Multiply by 16

[12]#color(white)("XXXX")##16x^2 +2500-300x+9x^2 = 1600#

Simplifying into standard quadratic form

[13]#color(white)("XXXX")##25x^2-300x+900 = 0#

Dividing by #25#

[14]#color(white)("XXXX")##x^2-12x+36 = 0#

Factoring

[15]#color(white)("XXXX")##(x-6)^2 = 0##color(white)("XXXX")##rarr x=6#

When #x=6# (substituting back into [8]

[16]#color(white)("XXXX")##y = (54-3(6))/4 = 9#