Solve {l}{2x-y+3z=25}{3x+2y-z=5}{5x+4y-6z=-11}

$$\left. \begin{array} { l } { 2 x - y + 3 z = 25 } \\ { 3 x + 2 y - z = 5 } \\ { 5 x + 4 y - 6 z = - 11 } \end{array} \right.$$

Solve for x, y, z

$$x=5$$

$$y=-3$$

$$z=4$$

Short Steps Using Substitution

Solve 2x-y+3z=25 for y.

$$y=2x+3z-25$$

Substitute 2x+3z-25 for y in the second and third equation.

$$3x+2\left(2x+3z-25\right)-z=5 5x+4\left(2x+3z-25\right)-6z=-11$$

Solve these equations for x and z respectively.

$$x=\frac{55}{7}-\frac{5}{7}z z=-\frac{13}{6}x+\frac{89}{6}$$

Substitute \frac{55}{7}-\frac{5}{7}z for x in the equation z=-\frac{13}{6}x+\frac{89}{6}.

$$z=-\frac{13}{6}\left(\frac{55}{7}-\frac{5}{7}z\right)+\frac{89}{6}$$

Solve z=-\frac{13}{6}\left(\frac{55}{7}-\frac{5}{7}z\right)+\frac{89}{6} for z.

$$z=4$$

Substitute 4 for z in the equation x=\frac{55}{7}-\frac{5}{7}z.

$$x=\frac{55}{7}-\frac{5}{7}\times 4$$

Calculate x from x=\frac{55}{7}-\frac{5}{7}\times 4.

$$x=5$$

Substitute 5 for x and 4 for z in the equation y=2x+3z-25.

$$y=2\times 5+3\times 4-25$$

Calculate y from y=2\times 5+3\times 4-25.

$$y=-3$$

The system is now solved.

$$x=5 y=-3 z=4$$