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

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

Solve for x, y, z

$$x=\frac{63}{125}=0.504$$

$$y=-\frac{11}{125}=-0.088$$

$$z=\frac{16}{125}=0.128$$

Short Steps Using Substitution

Reorder the equations.

$$4x-y+7z=3 5x+3y-2z=2 2x+3y+2z=1$$

Solve 4x-y+7z=3 for y.

$$y=4x+7z-3$$

Substitute 4x+7z-3 for y in the second and third equation.

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

Solve these equations for x and z respectively.

$$x=-\frac{19}{17}z+\frac{11}{17} z=\frac{10}{23}-\frac{14}{23}x$$

Substitute -\frac{19}{17}z+\frac{11}{17} for x in the equation z=\frac{10}{23}-\frac{14}{23}x.

$$z=\frac{10}{23}-\frac{14}{23}\left(-\frac{19}{17}z+\frac{11}{17}\right)$$

Solve z=\frac{10}{23}-\frac{14}{23}\left(-\frac{19}{17}z+\frac{11}{17}\right) for z.

$$z=\frac{16}{125}$$

Substitute \frac{16}{125} for z in the equation x=-\frac{19}{17}z+\frac{11}{17}.

$$x=-\frac{19}{17}\times \frac{16}{125}+\frac{11}{17}$$

Calculate x from x=-\frac{19}{17}\times \frac{16}{125}+\frac{11}{17}.

$$x=\frac{63}{125}$$

Substitute \frac{63}{125} for x and \frac{16}{125} for z in the equation y=4x+7z-3.

$$y=4\times \frac{63}{125}+7\times \frac{16}{125}-3$$

Calculate y from y=4\times \frac{63}{125}+7\times \frac{16}{125}-3.

$$y=-\frac{11}{125}$$

The system is now solved.

$$x=\frac{63}{125} y=-\frac{11}{125} z=\frac{16}{125}$$