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

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

Solve for x, y, z

$$x = \frac{13}{7} = 1\frac{6}{7} \approx 1.857142857$$

$$y=-\frac{2}{3}\approx -0.666666667$$

$$z=\frac{2}{21}\approx 0.095238095$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for y and z respectively.

$$y=-2+14z z=\frac{2}{7}+\frac{2}{7}y$$

Substitute -2+14z for y in the equation z=\frac{2}{7}+\frac{2}{7}y.

$$z=\frac{2}{7}+\frac{2}{7}\left(-2+14z\right)$$

Solve z=\frac{2}{7}+\frac{2}{7}\left(-2+14z\right) for z.

$$z=\frac{2}{21}$$

Substitute \frac{2}{21} for z in the equation y=-2+14z.

$$y=-2+14\times \frac{2}{21}$$

Calculate y from y=-2+14\times \frac{2}{21}.

$$y=-\frac{2}{3}$$

Substitute -\frac{2}{3} for y and \frac{2}{21} for z in the equation x=y-5z+3.

$$x=-\frac{2}{3}-5\times \frac{2}{21}+3$$

Calculate x from x=-\frac{2}{3}-5\times \frac{2}{21}+3.

$$x=\frac{13}{7}$$

The system is now solved.

$$x=\frac{13}{7} y=-\frac{2}{3} z=\frac{2}{21}$$