Solve {l}{2x_{1}+5x_{2}-3x_{3}+x_{4}=3}{3x_{1}+4x_{2}quad-2x_{4}=4}{3x_{1}+x_{2}-3x_{3}+6x_{4}=9}

$$\left. \begin{array} { l } { 2 x _ { 1 } + 5 x _ { 2 } - 3 x _ { 3 } + x _ { 4 } = 3 } \\ { 3 x _ { 1 } + 4 x _ { 2 } \quad - 2 x _ { 4 } = 4 } \\ { 3 x _ { 1 } + x _ { 2 } - 3 x _ { 3 } + 6 x _ { 4 } = 9 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=-\frac{3x_{4}}{4}+\frac{5}{2}$$

$$x_{2}=\frac{17x_{4}}{16}-\frac{7}{8}$$

$$x_{3}=\frac{77x_{4}}{48}-\frac{19}{24}$$

Short Steps Using Substitution

Reorder the equations.

$$3x_{1}+x_{2}-3x_{3}+6x_{4}=9 3x_{1}+4x_{2}-2x_{4}=4 2x_{1}+5x_{2}-3x_{3}+x_{4}=3$$

Solve 3x_{1}+x_{2}-3x_{3}+6x_{4}=9 for x_{2}.

$$x_{2}=9-3x_{1}+3x_{3}-6x_{4}$$

Substitute 9-3x_{1}+3x_{3}-6x_{4} for x_{2} in the second and third equation.

$$3x_{1}+4\left(9-3x_{1}+3x_{3}-6x_{4}\right)-2x_{4}=4 2x_{1}+5\left(9-3x_{1}+3x_{3}-6x_{4}\right)-3x_{3}+x_{4}=3$$

Solve these equations for x_{1} and x_{3} respectively.

$$x_{1}=\frac{32}{9}+\frac{4}{3}x_{3}-\frac{26}{9}x_{4} x_{3}=\frac{13}{12}x_{1}-\frac{7}{2}+\frac{29}{12}x_{4}$$

Substitute \frac{32}{9}+\frac{4}{3}x_{3}-\frac{26}{9}x_{4} for x_{1} in the equation x_{3}=\frac{13}{12}x_{1}-\frac{7}{2}+\frac{29}{12}x_{4}.

$$x_{3}=\frac{13}{12}\left(\frac{32}{9}+\frac{4}{3}x_{3}-\frac{26}{9}x_{4}\right)-\frac{7}{2}+\frac{29}{12}x_{4}$$

Solve x_{3}=\frac{13}{12}\left(\frac{32}{9}+\frac{4}{3}x_{3}-\frac{26}{9}x_{4}\right)-\frac{7}{2}+\frac{29}{12}x_{4} for x_{3}.

$$x_{3}=-\frac{19}{24}+\frac{77}{48}x_{4}$$

Substitute -\frac{19}{24}+\frac{77}{48}x_{4} for x_{3} in the equation x_{1}=\frac{32}{9}+\frac{4}{3}x_{3}-\frac{26}{9}x_{4}.

$$x_{1}=\frac{32}{9}+\frac{4}{3}\left(-\frac{19}{24}+\frac{77}{48}x_{4}\right)-\frac{26}{9}x_{4}$$

Calculate x_{1} from x_{1}=\frac{32}{9}+\frac{4}{3}\left(-\frac{19}{24}+\frac{77}{48}x_{4}\right)-\frac{26}{9}x_{4}.

$$x_{1}=\frac{5}{2}-\frac{3}{4}x_{4}$$

Substitute \frac{5}{2}-\frac{3}{4}x_{4} for x_{1} and -\frac{19}{24}+\frac{77}{48}x_{4} for x_{3} in the equation x_{2}=9-3x_{1}+3x_{3}-6x_{4}.

$$x_{2}=9-3\left(\frac{5}{2}-\frac{3}{4}x_{4}\right)+3\left(-\frac{19}{24}+\frac{77}{48}x_{4}\right)-6x_{4}$$

Calculate x_{2} from x_{2}=9-3\left(\frac{5}{2}-\frac{3}{4}x_{4}\right)+3\left(-\frac{19}{24}+\frac{77}{48}x_{4}\right)-6x_{4}.

$$x_{2}=-\frac{7}{8}+\frac{17}{16}x_{4}$$

The system is now solved.

$$x_{1}=\frac{5}{2}-\frac{3}{4}x_{4} x_{2}=-\frac{7}{8}+\frac{17}{16}x_{4} x_{3}=-\frac{19}{24}+\frac{77}{48}x_{4}$$