Solve {l}{-7x-y+5z=0}{4x-3y+4z=10}{-3x+4y+7t=42}

$$\left. \begin{array} { l } { - 7 x - y + 5 z = 0 } \\ { 4 x - 3 y + 4 z = 10 } \\ { - 3 x + 4 y + 7 t = 42 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{662-77t}{159}$$

$$y=\frac{722-112t}{53}$$

$$z=\frac{1360-175t}{159}$$

Short Steps Using Substitution

Solve -7x-y+5z=0 for y.

$$y=-7x+5z$$

Substitute -7x+5z for y in the second and third equation.

$$4x-3\left(-7x+5z\right)+4z=10 -3x+4\left(-7x+5z\right)+7t=42$$

Solve these equations for x and z respectively.

$$x=\frac{11}{25}z+\frac{2}{5} z=\frac{21}{10}+\frac{31}{20}x-\frac{7}{20}t$$

Substitute \frac{11}{25}z+\frac{2}{5} for x in the equation z=\frac{21}{10}+\frac{31}{20}x-\frac{7}{20}t.

$$z=\frac{21}{10}+\frac{31}{20}\left(\frac{11}{25}z+\frac{2}{5}\right)-\frac{7}{20}t$$

Solve z=\frac{21}{10}+\frac{31}{20}\left(\frac{11}{25}z+\frac{2}{5}\right)-\frac{7}{20}t for z.

$$z=\frac{1360}{159}-\frac{175}{159}t$$

Substitute \frac{1360}{159}-\frac{175}{159}t for z in the equation x=\frac{11}{25}z+\frac{2}{5}.

$$x=\frac{11}{25}\left(\frac{1360}{159}-\frac{175}{159}t\right)+\frac{2}{5}$$

Calculate x from x=\frac{11}{25}\left(\frac{1360}{159}-\frac{175}{159}t\right)+\frac{2}{5}.

$$x=\frac{662}{159}-\frac{77}{159}t$$

Substitute \frac{662}{159}-\frac{77}{159}t for x and \frac{1360}{159}-\frac{175}{159}t for z in the equation y=-7x+5z.

$$y=-7\left(\frac{662}{159}-\frac{77}{159}t\right)+5\left(\frac{1360}{159}-\frac{175}{159}t\right)$$

Calculate y from y=-7\left(\frac{662}{159}-\frac{77}{159}t\right)+5\left(\frac{1360}{159}-\frac{175}{159}t\right).

$$y=\frac{722}{53}-\frac{112}{53}t$$

The system is now solved.

$$x=\frac{662}{159}-\frac{77}{159}t y=\frac{722}{53}-\frac{112}{53}t z=\frac{1360}{159}-\frac{175}{159}t$$