\begin{cases} x + y = 0\\ \displaystyle\left(\frac{1+\sqrt{5}}{2}\right)x + \left(\frac{1 - \sqrt{5}}{2}\right)y = 1 \end{cases}

How can I solve for $x$ and $y$?

I can't seem to solve this so that when I plug the values back in, they satisfy both equations. Seems like elementary math but...does anyone have any tips on how to solve this?

  • $\begingroup$ Do you happen to know how to use a coefficient matrix to solve a system of linear equations? $\endgroup$ – 高田航 Oct 8 '18 at 0:47
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    $\begingroup$ Let $x=-y$, therefore $1=\frac {1+\sqrt 5}2x+\frac {-1+\sqrt 5}2x \implies \sqrt 5x=1 \implies x=\frac {\sqrt 5}5$ $\endgroup$ – Mohammad Zuhair Khan Oct 8 '18 at 0:48
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    $\begingroup$ It would help a lot if you showed us what you have so far. We are much more willing to show you what you did wrong than we are to tell you what the answer is. $\endgroup$ – steven gregory Oct 8 '18 at 1:07

Let $\phi=\frac{1+\sqrt{5}}{2}$, $\psi=\frac{1-\sqrt{5}}{2}$. Thus the system becomes $$S=\{(x,y)\in \Bbb R^2|(x+y=0),(\phi x+\psi y=1) \}$$ To find this, do the following

Given: $$x+y=0$$ $$\phi x+\psi y=1$$ Then: $$y=-x$$ $$\phi x+\psi y-1=0$$ Note: $$0=0$$ Therefore: $$ \phi x+\psi y-1=x+y$$ $$\phi x-x +\psi y-y=1$$ Recall: $y=-x$

Therefore: $$-x(\psi-1)+x(\phi-1)=1$$ $$x(\phi-1+1-\psi)=1$$ $$x=\frac{1}{\phi-\psi}$$ $$x=\frac{\sqrt{5}}{5}$$ $$y=-\frac{\sqrt{5}}{5}$$ $$S=\{(\frac{\sqrt{5}}{5},-\frac{\sqrt{5}}{5})\}$$


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