Solve {c}{(U_{1}-U_{2})1/2+1/10U_{1}+5I_{x}=2}{1/4U_{2}+(U_{2}-U_{1})1/2+5I_{x}=0}{I_{x}=frac{1}{3}}

$$\left. \begin{array} { c } { ( U _ { 1 } - U _ { 2 } ) \frac { 1 } { 2 } + \frac { 1 } { 10 } U _ { 1 } + 5 I _ { x } = 2 } \\ { \frac { 1 } { 4 } U _ { 2 } + ( U _ { 2 } - U _ { 1 } ) \frac { 1 } { 2 } + 5 I _ { x } = 0 } \\ { I _ { x } = \frac { 1 } { 3 } } \end{array} \right.$$

Solve for U_1, U_2, I_x

$$U_{1} = -\frac{35}{12} = -2\frac{11}{12} \approx -2.916666667$$

$$U_{2} = -\frac{25}{6} = -4\frac{1}{6} \approx -4.166666667$$

$$I_{x}=\frac{1}{3}\approx 0.333333333$$

Short Steps Using Substitution

Reorder the equations.

$$I_{x}=\frac{1}{3} \frac{1}{4}U_{2}+\left(U_{2}-U_{1}\right)\times \frac{1}{2}+5I_{x}=0 \left(U_{1}-U_{2}\right)\times \frac{1}{2}+\frac{1}{10}U_{1}+5I_{x}=2$$

Substitute \frac{1}{3} for I_{x} in the second and third equation.

$$\frac{1}{4}U_{2}+\left(U_{2}-U_{1}\right)\times \frac{1}{2}+5\times \frac{1}{3}=0 \left(U_{1}-U_{2}\right)\times \frac{1}{2}+\frac{1}{10}U_{1}+5\times \frac{1}{3}=2$$

Solve these equations for U_{2} and U_{1} respectively.

$$U_{2}=-\frac{20}{9}+\frac{2}{3}U_{1} U_{1}=\frac{5}{9}+\frac{5}{6}U_{2}$$

Substitute -\frac{20}{9}+\frac{2}{3}U_{1} for U_{2} in the equation U_{1}=\frac{5}{9}+\frac{5}{6}U_{2}.

$$U_{1}=\frac{5}{9}+\frac{5}{6}\left(-\frac{20}{9}+\frac{2}{3}U_{1}\right)$$

Solve U_{1}=\frac{5}{9}+\frac{5}{6}\left(-\frac{20}{9}+\frac{2}{3}U_{1}\right) for U_{1}.

$$U_{1}=-\frac{35}{12}$$

Substitute -\frac{35}{12} for U_{1} in the equation U_{2}=-\frac{20}{9}+\frac{2}{3}U_{1}.

$$U_{2}=-\frac{20}{9}+\frac{2}{3}\left(-\frac{35}{12}\right)$$

Calculate U_{2} from U_{2}=-\frac{20}{9}+\frac{2}{3}\left(-\frac{35}{12}\right).

$$U_{2}=-\frac{25}{6}$$

The system is now solved.

$$U_{1}=-\frac{35}{12} U_{2}=-\frac{25}{6} I_{x}=\frac{1}{3}$$