Solve three simultaneous equations with 3 unknowns

Question

(b) An electrical circuit comprises three closed loops giving the following equations for the currents $i_1, i_2$ and $i_3$

\begin{align*} i_1 + 8i_2 + 3i_3 &= -31\\ 3i_1 - 2i_2 + i_3 &= -5\\ 2i_1 - 3i_2 + 2i_3 &= 6 \end{align*}

This is the system I need to solve. How do I solve for all three?

Any help would be of great help. But I need step by step instructions for each unknown. Thanks

Answer 1

Hint : Multiply the first equation with $3$ and subtract the second to get one equation containing only $i_2$ and $i_3$. Multiply the first equation with $2$ and subtract the third to get another equation containing only $i_2$ and $i_3$. The result is

$$26i_2+8i_3=-88$$ $$19i_2+4i_3=-68$$

Now multiply the second of these equations with two and subtract the first.

You get $12i_2=-48$ , so $i_2=-4$. Use one of the two intermediate equations to get $i_3$ and finally calculate $i_1$ using one of the original equations.

The final result is $i_1=-5$ , $i_2=-4$ , $i_3=2$.

Answer 2

Isolate one of the variables (e.g. $i_1$) in the first equation, and substitute the result in the second, which becomes an equation with two unknowns ($i_2$ and $i_3$). Isolate one of them (e.g. $i_2$). Then substitute both results in the third equation, which (if you do it correctly) becomes an equation with only one unknown ($i_3$). Hopefully you know to solve that, and then you just put the result into the expression you found for $i_2$, and when you have that, put both values into the expression for $i_1$.