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Solve the following equations:
$ x + \sqrt {xy} + y = 65 \\
  {x^2} + xy + {y^2} = 2275 \\ $
This question has multiple correct options
${\text{A}}{\text{. }}x = 45,y = 5 \\
  {\text{B}}{\text{. }}x = 40,y = 15 \\
  {\text{C}}{\text{. }}x = 20,y = 5 \\
  {\text{D}}{\text{. }}x = 5,y = 45 \\ $

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Last updated date: 17th Jun 2024
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Answer
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Hint: In this type of problem first you must make sure that there is no present square root and cube root of any variables. If available we try to remove it by applying the algebraic formula.

Complete step-by-step solution:
Here in the question the given equations are
$x + \sqrt {xy} + y = 65$
$x + y = 65 - \sqrt {xy} $……......… let it be equation (1).
${x^2} + xy + {y^2} = 2275$….....… let it be equation (2).
Using ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
Here, on adding xy both side in equation(2) We can write it as:
${\left( {x + y} \right)^2} - xy = 2275$….. Let it be equation (3).
Substituting equation (1) in equation (2), we have
$ \Rightarrow {\left( {65 - \sqrt {xy} } \right)^2} - xy = 2275 \\
   \Rightarrow 4225 + xy - 130\sqrt {xy} - xy = 2275 \\
   \Rightarrow - 130\sqrt {xy} = - 1950 \\
   \Rightarrow \sqrt {xy} = 15 \\
   \Rightarrow xy = 225 \\
   \Rightarrow y = \dfrac{{225}}{x} \\ $
$\therefore y = \dfrac{{225}}{x}$
Now put the value of y in equation (2) we get,
$\Rightarrow {x^2} + {\left( {\dfrac{{225}}{x}} \right)^2} + x \cdot \dfrac{{225}}{x} = 2275 \\
   \Rightarrow {x^2} + \dfrac{{50652}}{{{x^2}}} = 2050 \\
   \Rightarrow {x^4} - 2050{x^2} + 50652 = 0 \\
   \Rightarrow {x^4} - 25{x^2} - 2025{x^2} + 50625 = 0 \\
   \Rightarrow {x^2}\left( {{x^2} - 25} \right) - 2025\left( {{x^2} - 25} \right) = 0 \\
   \Rightarrow \left( {{x^2} - 2025} \right)\left( {{x^2} - 25} \right) = 0 \\
   \Rightarrow {x^2} = 25,2025 \\
  \therefore x = \pm 5, \pm 45 $
Negative values of x are not satisfied by (1)
So, $x = 5, 45$
Now put the value of x in $y = \dfrac{{225}}{x}$ to find the value of y.
When $x=5, \Rightarrow y = \dfrac{{225}}{5} = 45$
When$ x=45, \Rightarrow y = \dfrac{{225}}{{45}} = 5$
Therefore, the values of x are 5, 45 and corresponding values of y are $45, 5.$
Hence options A and D are the correct answer.

Note: Whenever you come to this type of problem when we solve equations, we should make sure that equations don’t contain the square root & cube root of any variable. If it is then converting it to a simple form for reducing the complexity of the calculations and after that solve the equations.