Question

Fie x,y,z apartine R, astfel incat 2x + y = 7 radical 2 si 2y + 3z=13 radical 2.
Calculati 4x + 4y + 3z.
Ofer 50 pcte !

Answer 1

$\rm 2x+y=7 \sqrt{2} \Rightarrow 2(2x+y)=14 \sqrt{2} \Leftrightarrow 4x+2y=14 \sqrt 2 \Rightarrow \\ \Rightarrow 2y=14 \sqrt{2}-4x. \\ \\ 2y+3z=13 \sqrt{2} \Leftrightarrow 14 \sqrt{2} -4x+3z=13 \sqrt{2} \Leftrightarrow -4x+3z=- \sqrt 2 \Leftrightarrow \\ \Leftrightarrow 4x-3z= \sqrt{2} \Rightarrow 4x= \sqrt {2} +3z.$

$\rm 4x+4y+3z= \\ =\sqrt{2}+3z+4y+3z= \\ = \sqrt{2}+4y+6z= \\ = \sqrt{2}+2(2y+3z)= \\ = \sqrt{2}+2 \cdot 13 \sqrt{2} = \\ = \sqrt2 +26 \sqrt{2}= \\ =\boxed{ \underline{\bold{27 \sqrt{2} }}}. \\ \\ Deci~\boxed{4x+4y+3z=27 \sqrt{2}}.$