Solve {l}{x+2y+z=3}{-x+3y-2z=1}{3x+4y-7z=10}

$$\left\{ \begin{array} { l } { x + 2 y + z = 3 } \\ { - x + 3 y - 2 z = 1 } \\ { 3 x + 4 y - 7 z = 10 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

$$y=\frac{3}{4}=0.75$$

$$z=-\frac{1}{4}=-0.25$$

Short Steps Using Substitution

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

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

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

$$-\left(-2y-z+3\right)+3y-2z=1 3\left(-2y-z+3\right)+4y-7z=10$$

Solve these equations for y and z respectively.

$$y=\frac{4}{5}+\frac{1}{5}z z=-\frac{1}{5}y-\frac{1}{10}$$

Substitute \frac{4}{5}+\frac{1}{5}z for y in the equation z=-\frac{1}{5}y-\frac{1}{10}.

$$z=-\frac{1}{5}\left(\frac{4}{5}+\frac{1}{5}z\right)-\frac{1}{10}$$

Solve z=-\frac{1}{5}\left(\frac{4}{5}+\frac{1}{5}z\right)-\frac{1}{10} for z.

$$z=-\frac{1}{4}$$

Substitute -\frac{1}{4} for z in the equation y=\frac{4}{5}+\frac{1}{5}z.

$$y=\frac{4}{5}+\frac{1}{5}\left(-\frac{1}{4}\right)$$

Calculate y from y=\frac{4}{5}+\frac{1}{5}\left(-\frac{1}{4}\right).

$$y=\frac{3}{4}$$

Substitute \frac{3}{4} for y and -\frac{1}{4} for z in the equation x=-2y-z+3.

$$x=-2\times \frac{3}{4}-\left(-\frac{1}{4}\right)+3$$

Calculate x from x=-2\times \frac{3}{4}-\left(-\frac{1}{4}\right)+3.

$$x=\frac{7}{4}$$

The system is now solved.

$$x=\frac{7}{4} y=\frac{3}{4} z=-\frac{1}{4}$$