Answers (1)
x + y = 7 ;; isolate x
x = 7 - y ;; square
x^2 = (7 - y)^2 ;; expand
x^2 = y^2 - 14y + 49
The second equation is x^2 = y^2 + 25. So we can build the equation:
y^2 + 25 = y^2 - 14y + 49 ;; y^2 just cancels out
25 = -14y + 49 ;; -25 , +14y
14y = 24
y = 24 / 14
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Now we do the same thing the other way round isolating y in the original equations:
x + y = 7 ;; isolate y
y = 7 - x ;; square
y^2 = (7 - x)^2 ;; expand
y^2 = x^2 - 14x + 49
The second equation has to be rearranged a bit:
x^2 = y^2 + 25 ;; -x^2 , -y^2
-y^2 = -x^2 + 25 ;; times (-1)
y^2 = x^2 - 25
In both equations there is now y^2 on the left side.
We can, therefore, equate the right sides:
x^2 - 25 = x^2 - 14x + 49 ;; As before, x^2 just cancels out
-25 = -14x + 49 ;; +25 , +14x
14x = 74
x = 74 / 14
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Let's check the results:
x + y = 7
74/14 + 24/14 = 98/14 = 7
x^2 = y^2 + 25
74^2 / 14^2 = 24^2 / 14^2 + 25 ;; - (24^2 / 14^2)
(74^2 - 24^2) / 14^2 = 25
(5476 - 576) / 196 = 25
4900 / 196 = 25
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You might be tempted to (let your calculator) work out e.g. x = 74/14 but you'll get infinite recurrences and end up with
x+y = 6,9999999999999999999999999857143
and something like
y = sqrt((74 / 14)^2 - 25) = 1,7142857142857142857142857142857... ,
which is close enough for all practical purposes but it looks ugly and lacks the precision you get, when you are patient.