How to solve complex Simultaneous Linear Equations

Question

Find $a$ and $b$

(1) $a[(1+\sqrt{5})/2] + b[(1-\sqrt{5})/2] = 1$

(2) $a[(1+\sqrt{5})/2]^2 + b[(1-\sqrt{5})/2]^2 = 2$

Is there a smart method to solve this Simultaneous Linear Equations ?

Thanks you guys!

Answer 1

Write $\varphi=\dfrac{1+\sqrt{5}}{2}$, so that $$ \frac{1-\sqrt{5}}{2}=1-\varphi $$ and your system becomes \begin{cases} \varphi a+(1-\varphi)b=1\\ \varphi^2 a+(1-\varphi)^2 b=2 \end{cases}

Multiply the first equation by $1-\phi$ and subtract; now $$ a(\varphi(1-\varphi)-\varphi^2)=1-\varphi-2 $$ shouldn't be difficult to solve.

Answer 2

Let $\alpha=\frac{1+\sqrt5}{2}, \beta=\frac{1-\sqrt5}{2}$. Since we have ${\alpha}^2=1+\alpha, {\beta}^2=1+\beta,$ we have $$a(1+\alpha)+b(1+\beta)=2.$$ Then, you can get $a+b=1$. And?