Solve {l}{3x+2y=7}{2y+4z=3}{3z+x=1}

$$\left. \begin{array} { l } { 3 x + 2 y = 7 } \\ { 2 y + 4 z = 3 } \\ { 3 z + x = 1 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{16}{13} = 1\frac{3}{13} \approx 1.230769231$$

$$y = \frac{43}{26} = 1\frac{17}{26} \approx 1.653846154$$

$$z=-\frac{1}{13}\approx -0.076923077$$

Short Steps Using Substitution

Reorder the equations.

$$3z+x=1 2y+4z=3 3x+2y=7$$

Solve 3z+x=1 for x.

$$x=-3z+1$$

Substitute -3z+1 for x in the equation 3x+2y=7.

$$3\left(-3z+1\right)+2y=7$$

Solve the second equation for y and the third equation for z.

$$y=-2z+\frac{3}{2} z=-\frac{4}{9}+\frac{2}{9}y$$

Substitute -2z+\frac{3}{2} for y in the equation z=-\frac{4}{9}+\frac{2}{9}y.

$$z=-\frac{4}{9}+\frac{2}{9}\left(-2z+\frac{3}{2}\right)$$

Solve z=-\frac{4}{9}+\frac{2}{9}\left(-2z+\frac{3}{2}\right) for z.

$$z=-\frac{1}{13}$$

Substitute -\frac{1}{13} for z in the equation y=-2z+\frac{3}{2}.

$$y=-2\left(-\frac{1}{13}\right)+\frac{3}{2}$$

Calculate y from y=-2\left(-\frac{1}{13}\right)+\frac{3}{2}.

$$y=\frac{43}{26}$$

Substitute -\frac{1}{13} for z in the equation x=-3z+1.

$$x=-3\left(-\frac{1}{13}\right)+1$$

Calculate x from x=-3\left(-\frac{1}{13}\right)+1.

$$x=\frac{16}{13}$$

The system is now solved.

$$x=\frac{16}{13} y=\frac{43}{26} z=-\frac{1}{13}$$