Consider the set of equations $$\tag{A} x^{\frac 1 2}+y = 11 $$ $$\tag{B} x +y^{\frac 1 2} = 7 $$ With mere inspection and guessing, the Solution Set is ${4,9}$ . However I cannot not find a method to solve them beyond that.
Let $x^{\frac 1 2}=t $ . Then $x=t^2$ and $y=11-t$.
Inserting the value of x and y in terms of t into equation B yields
$t^4-13t+38=0 ……(C)$
Is there a mistake and what should be the next step to arrive at the solution. Equation $C$ is wrong as it has been pointed out.
We have $$y^{1/2}=7-x$$ so $$y=(7-x)^2$$ and $$x^{1/2}=11-(7-x)^2$$ and $$x-(14x-x^2-38)^2=0$$ this is $$- \left( x-4 \right) \left( {x}^{3}-24\,{x}^{2}+176\,x-361 \right) =0$$ after expanding and factorizing.