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

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

Solve for x, y, z

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$y=\frac{7}{9}\approx 0.777777778$$

$$z=\frac{7}{9}\approx 0.777777778$$

Short Steps Using Substitution

Solve 3x-y+z=5 for y.

$$y=3x+z-5$$

Substitute 3x+z-5 for y in the second and third equation.

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

Solve these equations for x and z respectively.

$$x=\frac{9}{4}-\frac{3}{4}z z=\frac{14}{3}-\frac{7}{3}x$$

Substitute \frac{9}{4}-\frac{3}{4}z for x in the equation z=\frac{14}{3}-\frac{7}{3}x.

$$z=\frac{14}{3}-\frac{7}{3}\left(\frac{9}{4}-\frac{3}{4}z\right)$$

Solve z=\frac{14}{3}-\frac{7}{3}\left(\frac{9}{4}-\frac{3}{4}z\right) for z.

$$z=\frac{7}{9}$$

Substitute \frac{7}{9} for z in the equation x=\frac{9}{4}-\frac{3}{4}z.

$$x=\frac{9}{4}-\frac{3}{4}\times \frac{7}{9}$$

Calculate x from x=\frac{9}{4}-\frac{3}{4}\times \frac{7}{9}.

$$x=\frac{5}{3}$$

Substitute \frac{5}{3} for x and \frac{7}{9} for z in the equation y=3x+z-5.

$$y=3\times \frac{5}{3}+\frac{7}{9}-5$$

Calculate y from y=3\times \frac{5}{3}+\frac{7}{9}-5.

$$y=\frac{7}{9}$$

The system is now solved.

$$x=\frac{5}{3} y=\frac{7}{9} z=\frac{7}{9}$$