Question

$\sqrt{x } + y = 11 \\ x + \sqrt{y } = 7 \\ find \: the \: value \: of \: x \: and \: y$

Answer 1

Dear Student.

THe following question is about the systematic equations in order to find the values of variable "x" and "y".

Now, Isolate the variable of "y" from the equation "_/x + y = 11":

$\bf{\sqrt{x} + y = 11}$

$\bf{\sqrt{x} + y - \sqrt{x} = 11 - \sqrt{x}}$

$\bf{y = 11 - \sqrt{x}}$

Since, y = 11 - _/x,  plug this value into the original equation of "x + _/y = 7":

$\bf{\therefore \quad x + \sqrt{11 - \sqrt{x}} = 7}$

Solve for the variable of "x" in this current equation:

$\bf{x + \sqrt{11 - \sqrt{x}} - x = 7 - x}$

$\bf{\sqrt{11 - \sqrt{x}} = 7 - x}$

$\bf{(\sqrt{11 - \sqrt{x}})^2 = (7 - x)^2}$

$\bf{((11 - \sqrt{x})^{\frac{1}{2}})^2 = 7^2 - 2 \times 7x + x^2}$

$\bf{(11 - \sqrt{x})^{\frac{1}{2} \times 2} = 49 - 14x + x^2}$

$\bf{11 - \sqrt{x} = 49 - 14x + x^}$

$\bf{11 - \sqrt{x} - 11 = 49 - 14x + x^2 - 11}$

$\bf{- \sqrt{x} = x^2 - 14x + 38}$

$\bf{(- \sqrt{x})^2 = (x^2 - 14x + 38)^2}$

$\bf{x = x^2x^2 + x^2 (- 14x) + x^2 \times 38 + (- 14x)x^2 + (- 14x)(- 14x) + (- 14x) \times 38 + 38}$

$\bf{x = x^4 - 28x^3 + 272x^2 - 1064x + 1444}$

$\bf{x^4 - 28x^3 + 272x^2 - 1065x + 1444 = 0}$

Solve this equation by the method of factoring:

$\bf{(x - 4)(x^3 - 24x^2 + 176x - 361) = 0}$

By Zero factor Principle:

$\bf{x - 4 = 0}$

$\bf{x = 4}$

No need to solve the second part, it's a more complex process and will be on approximation, therefore the answer is remaining as "x = 4", by verifying the solution:

Since,  x =4,

$\bf{4 + \sqrt{11 - \sqrt{4}} = 7}$

$\bf{4 + 3 = 7}$

$\bf{Therefore, \: the \: condition \: for \: x = 4 \: is \: true}$

Similarly, for x = 4, just plug in the following value into our first equation of "_/x + y = 11":

$\bf{\therefore \quad \sqrt{4} + y = 11}$

$\bf{\therefore \quad 2 + y = 11}$

$\boxed{\bf{\underline{\therefore \quad y = 9}}}$

Verify the solution for variable "y" to obtain the final solutions (Original Equations):

$\bf{\sqrt{4} + 9 = 11}$

$\bf{2 + 9 = 11}$

$\bf{11 = 11}$

THEREFORE, OUR FINAL SOLUTIONS FOR VARIABLE "x" AND "y" ARE:

$\boxed{\bf{\underline{x = 4}}}$

$\boxed{\bf{\underline{y = 9}}}$

Which is the final solution and the final answer for this type of query.

Hope it helps you and clears your doubts for obtaining the values for the given variables by a detailed process !!!

Answer 2

Hey !!! buddy UR solution... BUT.so...... wahiyaat. Question..........