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

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

Solve for x, y, z

$$x = \frac{186}{7} = 26\frac{4}{7} \approx 26.571428571$$

$$y = -\frac{229}{7} = -32\frac{5}{7} \approx -32.714285714$$

$$z = \frac{43}{7} = 6\frac{1}{7} \approx 6.142857143$$

Short Steps Using Substitution

Reorder the equations.

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

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

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

Solve these equations for y and z respectively.

$$y=41-12z z=-\frac{2}{5}-\frac{1}{5}y$$

Substitute 41-12z for y in the equation z=-\frac{2}{5}-\frac{1}{5}y.

$$z=-\frac{2}{5}-\frac{1}{5}\left(41-12z\right)$$

Solve z=-\frac{2}{5}-\frac{1}{5}\left(41-12z\right) for z.

$$z=\frac{43}{7}$$

Substitute \frac{43}{7} for z in the equation y=41-12z.

$$y=41-12\times \frac{43}{7}$$

Calculate y from y=41-12\times \frac{43}{7}.

$$y=-\frac{229}{7}$$

Substitute \frac{43}{7} for z in the equation x=4z+2.

$$x=4\times \frac{43}{7}+2$$

Calculate x from x=4\times \frac{43}{7}+2.

$$x=\frac{186}{7}$$

The system is now solved.

$$x=\frac{186}{7} y=-\frac{229}{7} z=\frac{43}{7}$$