Question 1092265
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<pre>
We are given these two equalities

x   +  y  = 11,      (1)
x^2 - y^2 = 33.      (2)


Factor (2) in this way:

x^2 - y^2 = (x+y)*(x-y).  


Then (2) takes the form

(x+y)*(x-y) = 33.    (3)


Now replace (x+y) in (3) by 11,  based on (1). You will get

11*(x-y) = 33,    or

x - y = 3.          (4)


Thus equalities (1) and (2), taken together, are equivalent to

x + y = 11,         (5)
x - y = 3.          (6)


Now add  (5) and (6) (both sides).  You will get

2x = 11+ 3 = 14.  Hence,  x = {{{14/2}}} = 7.


Then from (5)  y = 11-x = 11 - 7 = 4.


<U>Answer</U>.  The two numbers are 7 and 4.
</pre>

Solved.


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One can solve it by another way, by substituting &nbsp;y = 11-x &nbsp;from &nbsp;(1) &nbsp;into &nbsp;(2) &nbsp;and solving a quadratic equation.


I prefer my way, &nbsp;since it allows to avoid quadratic equation.



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The plot below is for illustration.



{{{graph( 630, 630, -20.5, 20.5, -10.5, 10.5,
          sqrt(x^2-33), -sqrt(x^2-33), 11-x
)}}}


Plot y = {{{sqrt(x^2-33)}}}  and  y = 11-x