Question 1105525

{{{ a^2 + b^2 = 65 }}}  (1)
{{{ (a+b)^2 = 121 }}}   (2)
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(2) —> {{{ (a+b) = 11 }}} —>   {{{ a = 11 - b }}} 

Substitute "11 - b" for "a" in (1):

{{{ (11-b)^2 + b^2 = 65 }}}
{{{  121 - 22b + b^2 + b^2 = 65 }}}
{{{ 2b^2 -22b +56 = 0 }}}
{{{ b^2 - 11b + 28 = 0 }}}
{{{  (b-7)(b-4) = 0 }}}
{{{ b=4 }}} or {{{ b=7 }}}

b=4  —>  a=7    
Check:   
{{{7^2 + 4^2  = 49+16 = 65 }}}  (ok)
 {{{ (4+7)^2 = 11^2 = 121 }}}  (ok)

b=7 —> a=4  (this is the same as above, the numbers are merely swapped)

—

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(2)  also —>  (a+b) = -11   —>   a = -11 - b

Substitute "-11 - b" for "a" in (1):

{{{ (-11-b)^2 + b^2 = 65 }}}
{{{  121 + 22b + b^2 + b^2 = 65 }}}
{{{ 2b^2 +22b +56 = 0 }}}
{{{ b^2 +11b + 28 = 0 }}}
{{{  (b+7)(b+4) = 0 }}}
{{{ b=-4 }}} or {{{ b=-7 }}}

b=-4  —> a=-11-(-4) = -7

—
Check:
  {{{ (-7)^2 + (-4)^2 = 49+16 = 65 }}}  (ok)
  {{{  (-4-7)^2 = (-11)^2  = 121  }}}   (ok)

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Answer (two solutions):
    (4 and 7)    and    
   (-4 and -7)

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