Solve {l}{x-y+z=22}{3x+y=47}{x=4z+2}

$$\left\{ \begin{array} { l } { x - y + z = 22 } \\ { 3 x + y = 47 } \\ { x = 4 z + 2 } \end{array} \right\}$$

Solve for x, y, z

$$x = \frac{278}{17} = 16\frac{6}{17} \approx 16.352941176$$

$$y = -\frac{35}{17} = -2\frac{1}{17} \approx -2.058823529$$

$$z = \frac{61}{17} = 3\frac{10}{17} \approx 3.588235294$$

Short Steps Using Substitution

Reorder the equations.

$$x=4z+2 3x+y=47 x-y+z=22$$

Substitute 2+4z for x in the second and third equation.

$$3\left(2+4z\right)+y=47 2+4z-y+z=22$$

Solve these equations for y and z respectively.

$$y=41-12z z=4+\frac{1}{5}y$$

Substitute 41-12z for y in the equation z=4+\frac{1}{5}y.

$$z=4+\frac{1}{5}\left(41-12z\right)$$

Solve z=4+\frac{1}{5}\left(41-12z\right) for z.

$$z=\frac{61}{17}$$

Substitute \frac{61}{17} for z in the equation y=41-12z.

$$y=41-12\times \frac{61}{17}$$

Calculate y from y=41-12\times \frac{61}{17}.

$$y=-\frac{35}{17}$$

Substitute \frac{61}{17} for z in the equation x=4z+2.

$$x=4\times \frac{61}{17}+2$$

Calculate x from x=4\times \frac{61}{17}+2.

$$x=\frac{278}{17}$$

The system is now solved.

$$x=\frac{278}{17} y=-\frac{35}{17} z=\frac{61}{17}$$