Solve {r}{3x-4y-z=11}{4x+2y+z=9}{7x-2y+3z=8}

$$\left. \begin{array} { r } { 3 x - 4 y - z = 11 } \\ { 4 x + 2 y + z = 9 } \\ { 7 x - 2 y + 3 z = 8 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=\frac{1}{2}=0.5$$

$$z=-4$$

Short Steps Using Substitution

Solve 3x-4y-z=11 for z.

$$z=3x-4y-11$$

Substitute 3x-4y-11 for z in the second and third equation.

$$4x+2y+3x-4y-11=9 7x-2y+3\left(3x-4y-11\right)=8$$

Solve these equations for y and x respectively.

$$y=\frac{7}{2}x-10 x=\frac{7}{8}y+\frac{41}{16}$$

Substitute \frac{7}{2}x-10 for y in the equation x=\frac{7}{8}y+\frac{41}{16}.

$$x=\frac{7}{8}\left(\frac{7}{2}x-10\right)+\frac{41}{16}$$

Solve x=\frac{7}{8}\left(\frac{7}{2}x-10\right)+\frac{41}{16} for x.

$$x=3$$

Substitute 3 for x in the equation y=\frac{7}{2}x-10.

$$y=\frac{7}{2}\times 3-10$$

Calculate y from y=\frac{7}{2}\times 3-10.

$$y=\frac{1}{2}$$

Substitute \frac{1}{2} for y and 3 for x in the equation z=3x-4y-11.

$$z=3\times 3-4\times \frac{1}{2}-11$$

Calculate z from z=3\times 3-4\times \frac{1}{2}-11.

$$z=-4$$

The system is now solved.

$$x=3 y=\frac{1}{2} z=-4$$