Solve {l}{2x+y+4z=16}{5x-2y+2z=-1}{x+2y-3z=-9}

$$\left. \begin{array} { l } { 2 x + y + 4 z = 16 } \\ { 5 x - 2 y + 2 z = - 1 } \\ { x + 2 y - 3 z = - 9 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=2$$

$$z=4$$

Short Steps Using Substitution

Solve 2x+y+4z=16 for y.

$$y=16-2x-4z$$

Substitute 16-2x-4z for y in the second and third equation.

$$5x-2\left(16-2x-4z\right)+2z=-1 x+2\left(16-2x-4z\right)-3z=-9$$

Solve these equations for x and z respectively.

$$x=\frac{31}{9}-\frac{10}{9}z z=-\frac{3}{11}x+\frac{41}{11}$$

Substitute \frac{31}{9}-\frac{10}{9}z for x in the equation z=-\frac{3}{11}x+\frac{41}{11}.

$$z=-\frac{3}{11}\left(\frac{31}{9}-\frac{10}{9}z\right)+\frac{41}{11}$$

Solve z=-\frac{3}{11}\left(\frac{31}{9}-\frac{10}{9}z\right)+\frac{41}{11} for z.

$$z=4$$

Substitute 4 for z in the equation x=\frac{31}{9}-\frac{10}{9}z.

$$x=\frac{31}{9}-\frac{10}{9}\times 4$$

Calculate x from x=\frac{31}{9}-\frac{10}{9}\times 4.

$$x=-1$$

Substitute -1 for x and 4 for z in the equation y=16-2x-4z.

$$y=16-2\left(-1\right)-4\times 4$$

Calculate y from y=16-2\left(-1\right)-4\times 4.

$$y=2$$

The system is now solved.

$$x=-1 y=2 z=4$$