Solve {l}{2x-3y+z=12}{3x+4y-5z=9}{-x+2y+3z=15}

$$\left. \begin{array} { l } { 2 x - 3 y + z = 12 } \\ { 3 x + 4 y - 5 z = 9 } \\ { - x + 2 y + 3 z = 15 } \end{array} \right.$$

Solve for x, y, z

$$x=8$$

$$y = \frac{35}{11} = 3\frac{2}{11} \approx 3.181818182$$

$$z = \frac{61}{11} = 5\frac{6}{11} \approx 5.545454545$$

Short Steps Using Substitution

Solve 2x-3y+z=12 for z.

$$z=-2x+3y+12$$

Substitute -2x+3y+12 for z in the second and third equation.

$$3x+4y-5\left(-2x+3y+12\right)=9 -x+2y+3\left(-2x+3y+12\right)=15$$

Solve these equations for y and x respectively.

$$y=\frac{13}{11}x-\frac{69}{11} x=\frac{11}{7}y+3$$

Substitute \frac{13}{11}x-\frac{69}{11} for y in the equation x=\frac{11}{7}y+3.

$$x=\frac{11}{7}\left(\frac{13}{11}x-\frac{69}{11}\right)+3$$

Solve x=\frac{11}{7}\left(\frac{13}{11}x-\frac{69}{11}\right)+3 for x.

$$x=8$$

Substitute 8 for x in the equation y=\frac{13}{11}x-\frac{69}{11}.

$$y=\frac{13}{11}\times 8-\frac{69}{11}$$

Calculate y from y=\frac{13}{11}\times 8-\frac{69}{11}.

$$y=\frac{35}{11}$$

Substitute \frac{35}{11} for y and 8 for x in the equation z=-2x+3y+12.

$$z=-2\times 8+3\times \frac{35}{11}+12$$

Calculate z from z=-2\times 8+3\times \frac{35}{11}+12.

$$z=\frac{61}{11}$$

The system is now solved.

$$x=8 y=\frac{35}{11} z=\frac{61}{11}$$