Solve {l}{3x+y+2z=1}{2x-3y+z=7}{4x+7y-3z=-5}

$$\left\{ \begin{array} { l } { 3 x + y + 2 z = 1 } \\ { 2 x - 3 y + z = 7 } \\ { 4 x + 7 y - 3 z = - 5 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{43}{34} = 1\frac{9}{34} \approx 1.264705882$$

$$y = -\frac{57}{34} = -1\frac{23}{34} \approx -1.676470588$$

$$z=-\frac{19}{34}\approx -0.558823529$$

Short Steps Using Substitution

Solve 3x+y+2z=1 for y.

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

Substitute -3x-2z+1 for y in the second and third equation.

$$2x-3\left(-3x-2z+1\right)+z=7 4x+7\left(-3x-2z+1\right)-3z=-5$$

Solve these equations for x and z respectively.

$$x=-\frac{7}{11}z+\frac{10}{11} z=-x+\frac{12}{17}$$

Substitute -\frac{7}{11}z+\frac{10}{11} for x in the equation z=-x+\frac{12}{17}.

$$z=-\left(-\frac{7}{11}z+\frac{10}{11}\right)+\frac{12}{17}$$

Solve z=-\left(-\frac{7}{11}z+\frac{10}{11}\right)+\frac{12}{17} for z.

$$z=-\frac{19}{34}$$

Substitute -\frac{19}{34} for z in the equation x=-\frac{7}{11}z+\frac{10}{11}.

$$x=-\frac{7}{11}\left(-\frac{19}{34}\right)+\frac{10}{11}$$

Calculate x from x=-\frac{7}{11}\left(-\frac{19}{34}\right)+\frac{10}{11}.

$$x=\frac{43}{34}$$

Substitute \frac{43}{34} for x and -\frac{19}{34} for z in the equation y=-3x-2z+1.

$$y=-3\times \frac{43}{34}-2\left(-\frac{19}{34}\right)+1$$

Calculate y from y=-3\times \frac{43}{34}-2\left(-\frac{19}{34}\right)+1.

$$y=-\frac{57}{34}$$

The system is now solved.

$$x=\frac{43}{34} y=-\frac{57}{34} z=-\frac{19}{34}$$