Solve {l}{3x+2y+4z=19E_{1}}{2x+3y+3z=21E_{z}}{3x-y+z=4quadE_{3}(-text{Elegem}}

$$\left. \begin{array} { l } { 3 x + 2 y + 4 z = 19 E _ { 1 } } \\ { 2 x + 3 y + 3 z = 21 E _ { z } } \\ { 3 x - y + z = 4 \quad E _ { 3 } ( - \text { Elegem } } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{21E_{z}}{2}-\frac{19E_{1}}{2}+2E_{3}$$

$$y=\frac{E_{3}}{3}+\frac{63E_{z}}{4}-\frac{133E_{1}}{12}$$

$$z=\frac{209E_{1}}{12}-\frac{5E_{3}}{3}-\frac{63E_{z}}{4}$$

Short Steps Using Substitution

Reorder the equations.

$$3x-y+z=4E_{3} 2x+3y+3z=21E_{z} 3x+2y+4z=19E_{1}$$

Solve 3x-y+z=4E_{3} for y.

$$y=3x+z-4E_{3}$$

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

$$2x+3\left(3x+z-4E_{3}\right)+3z=21E_{z} 3x+2\left(3x+z-4E_{3}\right)+4z=19E_{1}$$

Solve these equations for x and z respectively.

$$x=-\frac{6}{11}z+\frac{12}{11}E_{3}+\frac{21}{11}E_{z} z=-\frac{3}{2}x+\frac{4}{3}E_{3}+\frac{19}{6}E_{1}$$

Substitute -\frac{6}{11}z+\frac{12}{11}E_{3}+\frac{21}{11}E_{z} for x in the equation z=-\frac{3}{2}x+\frac{4}{3}E_{3}+\frac{19}{6}E_{1}.

$$z=-\frac{3}{2}\left(-\frac{6}{11}z+\frac{12}{11}E_{3}+\frac{21}{11}E_{z}\right)+\frac{4}{3}E_{3}+\frac{19}{6}E_{1}$$

Solve z=-\frac{3}{2}\left(-\frac{6}{11}z+\frac{12}{11}E_{3}+\frac{21}{11}E_{z}\right)+\frac{4}{3}E_{3}+\frac{19}{6}E_{1} for z.

$$z=\frac{209}{12}E_{1}-\frac{63}{4}E_{z}-\frac{5}{3}E_{3}$$

Substitute \frac{209}{12}E_{1}-\frac{63}{4}E_{z}-\frac{5}{3}E_{3} for z in the equation x=-\frac{6}{11}z+\frac{12}{11}E_{3}+\frac{21}{11}E_{z}.

$$x=-\frac{6}{11}\left(\frac{209}{12}E_{1}-\frac{63}{4}E_{z}-\frac{5}{3}E_{3}\right)+\frac{12}{11}E_{3}+\frac{21}{11}E_{z}$$

Calculate x from x=-\frac{6}{11}\left(\frac{209}{12}E_{1}-\frac{63}{4}E_{z}-\frac{5}{3}E_{3}\right)+\frac{12}{11}E_{3}+\frac{21}{11}E_{z}.

$$x=-\frac{19}{2}E_{1}+\frac{21}{2}E_{z}+2E_{3}$$

Substitute -\frac{19}{2}E_{1}+\frac{21}{2}E_{z}+2E_{3} for x and \frac{209}{12}E_{1}-\frac{63}{4}E_{z}-\frac{5}{3}E_{3} for z in the equation y=3x+z-4E_{3}.

$$y=3\left(-\frac{19}{2}E_{1}+\frac{21}{2}E_{z}+2E_{3}\right)+\frac{209}{12}E_{1}-\frac{63}{4}E_{z}-\frac{5}{3}E_{3}-4E_{3}$$

Calculate y from y=3\left(-\frac{19}{2}E_{1}+\frac{21}{2}E_{z}+2E_{3}\right)+\frac{209}{12}E_{1}-\frac{63}{4}E_{z}-\frac{5}{3}E_{3}-4E_{3}.

$$y=-\frac{133}{12}E_{1}+\frac{63}{4}E_{z}+\frac{1}{3}E_{3}$$

The system is now solved.

$$x=-\frac{19}{2}E_{1}+\frac{21}{2}E_{z}+2E_{3} y=-\frac{133}{12}E_{1}+\frac{63}{4}E_{z}+\frac{1}{3}E_{3} z=\frac{209}{12}E_{1}-\frac{63}{4}E_{z}-\frac{5}{3}E_{3}$$