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

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

Solve for x, y, z

$$x=-\frac{2}{47}\approx -0.042553191$$

$$y = \frac{51}{47} = 1\frac{4}{47} \approx 1.085106383$$

$$z=\frac{7}{47}\approx 0.14893617$$

Short Steps Using Substitution

Solve 3x+y+7z=2 for y.

$$y=-3x-7z+2$$

Substitute -3x-7z+2 for y in the second and third equation.

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

Solve these equations for x and z respectively.

$$x=-7z+1 z=-\frac{9}{16}x+\frac{1}{8}$$

Substitute -7z+1 for x in the equation z=-\frac{9}{16}x+\frac{1}{8}.

$$z=-\frac{9}{16}\left(-7z+1\right)+\frac{1}{8}$$

Solve z=-\frac{9}{16}\left(-7z+1\right)+\frac{1}{8} for z.

$$z=\frac{7}{47}$$

Substitute \frac{7}{47} for z in the equation x=-7z+1.

$$x=-7\times \frac{7}{47}+1$$

Calculate x from x=-7\times \frac{7}{47}+1.

$$x=-\frac{2}{47}$$

Substitute -\frac{2}{47} for x and \frac{7}{47} for z in the equation y=-3x-7z+2.

$$y=-3\left(-\frac{2}{47}\right)-7\times \frac{7}{47}+2$$

Calculate y from y=-3\left(-\frac{2}{47}\right)-7\times \frac{7}{47}+2.

$$y=\frac{51}{47}$$

The system is now solved.

$$x=-\frac{2}{47} y=\frac{51}{47} z=\frac{7}{47}$$