Solve the following parametric system of three equations and three unknowns This is my first time posting here so I'm exciting to join the community and gain as much knowledge as possible. My algebra is quite lacking and I'm a first year Physicist. Please could you help me solve the following three equations to find $I_3$ in terms of only $R$s and $\xi$ (i.e. no $I_1$ or $I_2$ terms). Thank you!
$$\begin{eqnarray}
\xi_1 - I_1 R_1 - (I_1 - I_2) R_4    &=& 0\\ 
- I_2 R_2 - (I_2 - I_3) R_5 - (I_2 - I_1) R_4  &=& 0\\
 - I_3 R_3- I_3 R_6 - (I_3 - I_2) R_5  &=& 0
\end{eqnarray}$$
 A: So far, I get "augmented matrix"
$$
\left(
\begin{array}{ccc|c}
R_1 + R_4 & - R_4 & 0 & \xi_1 \\
-R_4 & R_2 + R_4 + R_5 & - R_5 & 0 \\
0 & - R_5 & R_3 + R_5 + R_6 & 0
\end{array}
\right)
$$
OOH, symmetric square part
A: I calculated the solution with Maxima

(%i1) list_of_equations:[xi1-I1*R1-(I1-I2)*R4=0, -I2*R2-(I2-I3)*R5-(I2-I1)*R4=0,-I3*R3-I3*R6-(I3-I2)*R5=0];
(%o1) [(I2 - I1) R4 - I1 R1 + xi1 = 0, 
 - (I2 - I3) R5 + (I1 - I2) R4 - I2 R2 = 0, - I3 R6 + (I2 - I3) R5 - I3 R3 = 0]
(%i2) list_of_unknowns:[I1,I2,I3];
(%o2)                            [I1, I2, I3]
(%i3) list_of_solutions:solve(list_of_equations,list_of_unknowns);
(%o3) [[I1 = (xi1 (R5 (R6 + R3 + R2) + R2 (R6 + R3)) + xi1 R4 (R6 + R5 + R3))
/(R4 (R1 (R6 + R5 + R3) + R5 (R6 + R3 + R2) + R2 (R6 + R3))
 + R1 (R5 (R6 + R3 + R2) + R2 (R6 + R3))), 
I2 = (xi1 R4 (R6 + R5 + R3))/(R4 (R1 (R6 + R5 + R3) + R5 (R6 + R3 + R2)
 + R2 (R6 + R3)) + R1 (R5 (R6 + R3 + R2) + R2 (R6 + R3))), 
I3 = (xi1 R4 R5)/(R4 (R1 (R6 + R5 + R3) + R5 (R6 + R3 + R2) + R2 (R6 + R3))
 + R1 (R5 (R6 + R3 + R2) + R2 (R6 + R3)))]]

