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

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

Solve for x, y, z

$$x=2$$

$$y=3$$

$$z=-1$$

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.

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

Solve these equations for x and z respectively.

$$x=-\frac{5}{3}z+\frac{1}{3} z=-\frac{9}{16}x+\frac{1}{8}$$

Substitute -\frac{5}{3}z+\frac{1}{3} for x in the equation z=-\frac{9}{16}x+\frac{1}{8}.

$$z=-\frac{9}{16}\left(-\frac{5}{3}z+\frac{1}{3}\right)+\frac{1}{8}$$

Solve z=-\frac{9}{16}\left(-\frac{5}{3}z+\frac{1}{3}\right)+\frac{1}{8} for z.

$$z=-1$$

Substitute -1 for z in the equation x=-\frac{5}{3}z+\frac{1}{3}.

$$x=-\frac{5}{3}\left(-1\right)+\frac{1}{3}$$

Calculate x from x=-\frac{5}{3}\left(-1\right)+\frac{1}{3}.

$$x=2$$

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

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

Calculate y from y=-3\times 2-7\left(-1\right)+2.

$$y=3$$

The system is now solved.

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