Question 188802
{{{ x^2  - 8y = 104}}} Start with the second equation.


{{{ x^2 = 8y+104}}} Add 8y to both sides.



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{{{ x^2 + y^2  = 169}}} Move onto the second equation



{{{ 8y+104 + y^2  = 169}}} Plug in {{{ x^2 = 8y+104}}}



{{{ 8y+104 + y^2  - 169=0}}} Subtract 169 from both sides.



{{{y^2+8y-65=0}}} Combine like terms.




Notice we have a quadratic equation in the form of {{{ay^2+by+c}}} where {{{a=1}}}, {{{b=8}}}, and {{{c=-65}}}



Let's use the quadratic formula to solve for y



{{{y = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{y = (-(8) +- sqrt( (8)^2-4(1)(-65) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=8}}}, and {{{c=-65}}}



{{{y = (-8 +- sqrt( 64-4(1)(-65) ))/(2(1))}}} Square {{{8}}} to get {{{64}}}. 



{{{y = (-8 +- sqrt( 64--260 ))/(2(1))}}} Multiply {{{4(1)(-65)}}} to get {{{-260}}}



{{{y = (-8 +- sqrt( 64+260 ))/(2(1))}}} Rewrite {{{sqrt(64--260)}}} as {{{sqrt(64+260)}}}



{{{y = (-8 +- sqrt( 324 ))/(2(1))}}} Add {{{64}}} to {{{260}}} to get {{{324}}}



{{{y = (-8 +- sqrt( 324 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{y = (-8 +- 18)/(2)}}} Take the square root of {{{324}}} to get {{{18}}}. 



{{{y = (-8 + 18)/(2)}}} or {{{y = (-8 - 18)/(2)}}} Break up the expression. 



{{{y = (10)/(2)}}} or {{{y =  (-26)/(2)}}} Combine like terms. 



{{{y = 5}}} or {{{y = -13}}} Simplify. 



So the answers for "y" are {{{y = 5}}} or {{{y = -13}}} 



Now simply plug these solutions into {{{x^2=8y+104}}} to find the solutions for "x". I'll let you do this.



Note: you should find that there are 3 ordered pair solutions.