Solve {l}{x+2y+3z=14}{5x-y-z=0}{4x+3y+2z=16}

$$\left\{ \begin{array} { l } { x + 2 y + 3 z = 14 } \\ { 5 x - y - z = 0 } \\ { 4 x + 3 y + 2 z = 16 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=3$$

Short Steps Using Substitution

Solve x+2y+3z=14 for x.

$$x=-2y-3z+14$$

Substitute -2y-3z+14 for x in the second and third equation.

$$5\left(-2y-3z+14\right)-y-z=0 4\left(-2y-3z+14\right)+3y+2z=16$$

Solve these equations for y and z respectively.

$$y=\frac{70}{11}-\frac{16}{11}z z=4-\frac{1}{2}y$$

Substitute \frac{70}{11}-\frac{16}{11}z for y in the equation z=4-\frac{1}{2}y.

$$z=4-\frac{1}{2}\left(\frac{70}{11}-\frac{16}{11}z\right)$$

Solve z=4-\frac{1}{2}\left(\frac{70}{11}-\frac{16}{11}z\right) for z.

$$z=3$$

Substitute 3 for z in the equation y=\frac{70}{11}-\frac{16}{11}z.

$$y=\frac{70}{11}-\frac{16}{11}\times 3$$

Calculate y from y=\frac{70}{11}-\frac{16}{11}\times 3.

$$y=2$$

Substitute 2 for y and 3 for z in the equation x=-2y-3z+14.

$$x=-2\times 2-3\times 3+14$$

Calculate x from x=-2\times 2-3\times 3+14.

$$x=1$$

The system is now solved.

$$x=1 y=2 z=3$$