I have the following system of four equations:
$$\begin{align} &x_1 = \dfrac{1}{2} x_2 + \dfrac{1}{4} x_3 \\ &x_2 = \dfrac{1}{3} x_1 + \dfrac{1}{2} x_4 \\ &x_3 = \dfrac{2}{3} x_1 + \dfrac{1}{2} x_4 \\ &x_1 + x_2 + x_3 + x_4 = 1 \end{align}$$
How do I solve this system to get a unique solution for $\mathbf{x}$? I tried doing it by hand by substituting the first equation into the second, and then the third into the second, and then everything into the last one, but that doesn't work. I could use an online calculator, but I'd like to know what I'm doing wrong and how it is done by hand. Is there a more systematic/orderly way to do this, such as with Gaussian elimination / row reduction?
If you want to use substitution.
Solve the first equation for $x_3$; plug it. Solve the second equation for $x_4$; plug it. Solve the thirs equation for $x_2$.
At this point, $x_1,x_2,x_3$ as expressed as functions of $x_1$. Plug in the last equation and solve for $x_1$ and go backward to get the other.