# HHL example solution on Qiskit

In the example on the Qiskit page (https://qiskit.org/textbook/ch-applications/hhl_tutorial.html#A.-Some-mathematical-background) section 3 for the HHL algorithm the author is trying to solve the system of equations written as $$A \vec{x}=\vec{b}$$, where: $$A=\left(\begin{array}{cc} 1 & -1 / 3 \\ -1 / 3 & 1 \end{array}\right), \quad \vec{b}=\left(\begin{array}{c} 1 \\ 0 \end{array}\right) \quad \text { and } \quad \vec{x}=\left(\begin{array}{c} x_{1} \\ x_{2} \end{array}\right).$$

The final state of the system implementing the algorithm is: $$\frac{\frac{3}{2 \sqrt{2}}\left|u_{1}\right\rangle+\frac{3}{4 \sqrt{2}}\left|u_{2}\right\rangle}{\sqrt{45 / 32}}=\frac{|x\rangle}{\|x\|}.$$ Given that: $$|x\rangle={\frac{3}{2 \sqrt{2}}\left|u_{1}\right\rangle+\frac{3}{4 \sqrt{2}}\left|u_{2}\right\rangle}$$

where: $$\left|u_{1}\right\rangle=\left(\begin{array}{c} 1 \\ -1 \end{array}\right) \quad \text { and } \quad\left|u_{2}\right\rangle=\left(\begin{array}{l} 1 \\ 1 \end{array}\right).$$

How do i determine the solution to the system which is $$\left[\begin{array}{ll}1.125 & 0.375\end{array}\right]$$?