Solve {r}{3x+4y+z=4}{6x-y+3z=-5}{5y+z=11}

$$\left. \begin{array} { r } { 3 x + 4 y + z = 4 } \\ { 6 x - y + 3 z = - 5 } \\ { 5 y + z = 11 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{37}{21} = -1\frac{16}{21} \approx -1.761904762$$

$$y = \frac{12}{7} = 1\frac{5}{7} \approx 1.714285714$$

$$z = \frac{17}{7} = 2\frac{3}{7} \approx 2.428571429$$

Short Steps Using Substitution

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

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

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

$$6x-y+3\left(-3x-4y+4\right)=-5 5y-3x-4y+4=11$$

Solve these equations for y and x respectively.

$$y=-\frac{3}{13}x+\frac{17}{13} x=-\frac{7}{3}+\frac{1}{3}y$$

Substitute -\frac{3}{13}x+\frac{17}{13} for y in the equation x=-\frac{7}{3}+\frac{1}{3}y.

$$x=-\frac{7}{3}+\frac{1}{3}\left(-\frac{3}{13}x+\frac{17}{13}\right)$$

Solve x=-\frac{7}{3}+\frac{1}{3}\left(-\frac{3}{13}x+\frac{17}{13}\right) for x.

$$x=-\frac{37}{21}$$

Substitute -\frac{37}{21} for x in the equation y=-\frac{3}{13}x+\frac{17}{13}.

$$y=-\frac{3}{13}\left(-\frac{37}{21}\right)+\frac{17}{13}$$

Calculate y from y=-\frac{3}{13}\left(-\frac{37}{21}\right)+\frac{17}{13}.

$$y=\frac{12}{7}$$

Substitute \frac{12}{7} for y and -\frac{37}{21} for x in the equation z=-3x-4y+4.

$$z=-3\left(-\frac{37}{21}\right)-4\times \frac{12}{7}+4$$

Calculate z from z=-3\left(-\frac{37}{21}\right)-4\times \frac{12}{7}+4.

$$z=\frac{17}{7}$$

The system is now solved.

$$x=-\frac{37}{21} y=\frac{12}{7} z=\frac{17}{7}$$