# 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!

$$\begin{eqnarray} \xi_1 - I_1 R_1 - (I_1 - I_2) R_4 &=& 0\\ - I_2 R_2 - (I_2 - I_3) R_5 - (I_2 - I_1) R_4 &=& 0\\ - I_3 R_3- I_3 R_6 - (I_3 - I_2) R_5 &=& 0 \end{eqnarray}$$

• this System should be solved for $$I_1,I_2,I_3$$? – Dr. Sonnhard Graubner Dec 27 '17 at 22:37
• Why don't you write out the linear system in $I_k$ and solve it? – copper.hat Dec 27 '17 at 22:38
• you really should negate everything, put the constant $\xi_1$ on the other side of the equals, and carefully write out the three by three linear system, where the "variables" are $I_1, I_2, I_3.$ Unless nonmaximal rank, there should be just one solution – Will Jagy Dec 27 '17 at 22:39
• $\xi$ is called Xi, not epsilon. – miracle173 Dec 27 '17 at 22:41
• The first equation gives you $I_2$ in terms of $I_1$. Replace in the other equations, and you get a system of just two equations with two unknowns $I_1, I_3\,$. Repeat once more to eliminate $I_3$, and you are down to one equation with the single unknown $I_1$. Solve that and you are done. – dxiv Dec 27 '17 at 22:43

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

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)))]]

• I am sorry but you don't help the OP by doing that. The OP needs guiding and method. You should at least make him the remark that the denominators are the same... – Jean Marie Dec 27 '17 at 23:04
• @JeanMarie I was only interested to check if a solution exists by using a cas. And decided to communicate this finding to the OP. To do this in a comment would be more appropriate but I see no way to format a comment usefully. – miracle173 Dec 28 '17 at 8:31