Solve {l}{3u-2y+4z=110}{2u+2y-3z=-30}{-2u+6z+4z=220}

$$\left. \begin{array} { l } { 3 u - 2 y + 4 z = 110 } \\ { 2 u + 2 y - 3 z = - 30 } \\ { - 2 u + 6 z + 4 z = 220 } \end{array} \right.$$

Solve for u, y, z

$$y = \frac{265}{26} = 10\frac{5}{26} \approx 10.192307692$$

$$z = \frac{315}{13} = 24\frac{3}{13} \approx 24.230769231$$

$$u = \frac{145}{13} = 11\frac{2}{13} \approx 11.153846154$$

Short Steps Using Substitution

Solve 3u-2y+4z=110 for u.

$$u=\frac{110}{3}+\frac{2}{3}y-\frac{4}{3}z$$

Substitute \frac{110}{3}+\frac{2}{3}y-\frac{4}{3}z for u in the second and third equation.

$$2\left(\frac{110}{3}+\frac{2}{3}y-\frac{4}{3}z\right)+2y-3z=-30 -2\left(\frac{110}{3}+\frac{2}{3}y-\frac{4}{3}z\right)+6z+4z=220$$

Solve these equations for y and z respectively.

$$y=-31+\frac{17}{10}z z=\frac{440}{19}+\frac{2}{19}y$$

Substitute -31+\frac{17}{10}z for y in the equation z=\frac{440}{19}+\frac{2}{19}y.

$$z=\frac{440}{19}+\frac{2}{19}\left(-31+\frac{17}{10}z\right)$$

Solve z=\frac{440}{19}+\frac{2}{19}\left(-31+\frac{17}{10}z\right) for z.

$$z=\frac{315}{13}$$

Substitute \frac{315}{13} for z in the equation y=-31+\frac{17}{10}z.

$$y=-31+\frac{17}{10}\times \frac{315}{13}$$

Calculate y from y=-31+\frac{17}{10}\times \frac{315}{13}.

$$y=\frac{265}{26}$$

Substitute \frac{265}{26} for y and \frac{315}{13} for z in the equation u=\frac{110}{3}+\frac{2}{3}y-\frac{4}{3}z.

$$u=\frac{110}{3}+\frac{2}{3}\times \frac{265}{26}-\frac{4}{3}\times \frac{315}{13}$$

Calculate u from u=\frac{110}{3}+\frac{2}{3}\times \frac{265}{26}-\frac{4}{3}\times \frac{315}{13}.

$$u=\frac{145}{13}$$

The system is now solved.

$$u=\frac{145}{13} y=\frac{265}{26} z=\frac{315}{13}$$