Solve {c}{5x-4y+z=8}{4x+5y+4z=-15}{3x-y+z=1}

$$\left. \begin{array} { c } { 5 x - 4 y + z = 8 } \\ { 4 x + 5 y + 4 z = - 15 } \\ { 3 x - y + z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=-3$$

$$z=1$$

Short Steps Using Substitution

Solve 5x-4y+z=8 for z.

$$z=-5x+4y+8$$

Substitute -5x+4y+8 for z in the second and third equation.

$$4x+5y+4\left(-5x+4y+8\right)=-15 3x-y-5x+4y+8=1$$

Solve these equations for y and x respectively.

$$y=\frac{16}{21}x-\frac{47}{21} x=\frac{3}{2}y+\frac{7}{2}$$

Substitute \frac{16}{21}x-\frac{47}{21} for y in the equation x=\frac{3}{2}y+\frac{7}{2}.

$$x=\frac{3}{2}\left(\frac{16}{21}x-\frac{47}{21}\right)+\frac{7}{2}$$

Solve x=\frac{3}{2}\left(\frac{16}{21}x-\frac{47}{21}\right)+\frac{7}{2} for x.

$$x=-1$$

Substitute -1 for x in the equation y=\frac{16}{21}x-\frac{47}{21}.

$$y=\frac{16}{21}\left(-1\right)-\frac{47}{21}$$

Calculate y from y=\frac{16}{21}\left(-1\right)-\frac{47}{21}.

$$y=-3$$

Substitute -3 for y and -1 for x in the equation z=-5x+4y+8.

$$z=-5\left(-1\right)+4\left(-3\right)+8$$

Calculate z from z=-5\left(-1\right)+4\left(-3\right)+8.

$$z=1$$

The system is now solved.

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