Solve {l}{2x+y-3z=1}{x-2y+4z=19}{3x+4y-z=1}

$$\left. \begin{array} { l } { 2 x + y - 3 z = 1 } \\ { x - 2 y + 4 z = 19 } \\ { 3 x + 4 y - z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=-3$$

$$z=2$$

Short Steps Using Substitution

Solve 2x+y-3z=1 for y.

$$y=-2x+3z+1$$

Substitute -2x+3z+1 for y in the second and third equation.

$$x-2\left(-2x+3z+1\right)+4z=19 3x+4\left(-2x+3z+1\right)-z=1$$

Solve these equations for x and z respectively.

$$x=\frac{2}{5}z+\frac{21}{5} z=\frac{5}{11}x-\frac{3}{11}$$

Substitute \frac{2}{5}z+\frac{21}{5} for x in the equation z=\frac{5}{11}x-\frac{3}{11}.

$$z=\frac{5}{11}\left(\frac{2}{5}z+\frac{21}{5}\right)-\frac{3}{11}$$

Solve z=\frac{5}{11}\left(\frac{2}{5}z+\frac{21}{5}\right)-\frac{3}{11} for z.

$$z=2$$

Substitute 2 for z in the equation x=\frac{2}{5}z+\frac{21}{5}.

$$x=\frac{2}{5}\times 2+\frac{21}{5}$$

Calculate x from x=\frac{2}{5}\times 2+\frac{21}{5}.

$$x=5$$

Substitute 5 for x and 2 for z in the equation y=-2x+3z+1.

$$y=-2\times 5+3\times 2+1$$

Calculate y from y=-2\times 5+3\times 2+1.

$$y=-3$$

The system is now solved.

$$x=5 y=-3 z=2$$