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

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

Solve for x, y, z

$$x = \frac{12}{11} = 1\frac{1}{11} \approx 1.090909091$$

$$y = \frac{29}{11} = 2\frac{7}{11} \approx 2.636363636$$

$$z=\frac{8}{11}\approx 0.727272727$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for x and z respectively.

$$x=\frac{1}{8}z+1 z=\frac{2}{3}x$$

Substitute \frac{1}{8}z+1 for x in the equation z=\frac{2}{3}x.

$$z=\frac{2}{3}\left(\frac{1}{8}z+1\right)$$

Solve z=\frac{2}{3}\left(\frac{1}{8}z+1\right) for z.

$$z=\frac{8}{11}$$

Substitute \frac{8}{11} for z in the equation x=\frac{1}{8}z+1.

$$x=\frac{1}{8}\times \frac{8}{11}+1$$

Calculate x from x=\frac{1}{8}\times \frac{8}{11}+1.

$$x=\frac{12}{11}$$

Substitute \frac{12}{11} for x and \frac{8}{11} for z in the equation y=-3x+4z+3.

$$y=-3\times \frac{12}{11}+4\times \frac{8}{11}+3$$

Calculate y from y=-3\times \frac{12}{11}+4\times \frac{8}{11}+3.

$$y=\frac{29}{11}$$

The system is now solved.

$$x=\frac{12}{11} y=\frac{29}{11} z=\frac{8}{11}$$