Solve {l}{x+2y+z=3}{2x-y+z=7}{x-5y+6z=15}

$$\left. \begin{array} { l } { x + 2 y + z = 3 } \\ { 2 x - y + z = 7 } \\ { x - 5 y + 6 z = 15 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{77}{32} = 2\frac{13}{32} = 2.40625$$

$$y=-\frac{17}{32}=-0.53125$$

$$z = \frac{53}{32} = 1\frac{21}{32} = 1.65625$$

Short Steps Using Substitution

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

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

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

$$2\left(-2y-z+3\right)-y+z=7 -2y-z+3-5y+6z=15$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{5}z-\frac{1}{5} z=\frac{7}{5}y+\frac{12}{5}$$

Substitute -\frac{1}{5}z-\frac{1}{5} for y in the equation z=\frac{7}{5}y+\frac{12}{5}.

$$z=\frac{7}{5}\left(-\frac{1}{5}z-\frac{1}{5}\right)+\frac{12}{5}$$

Solve z=\frac{7}{5}\left(-\frac{1}{5}z-\frac{1}{5}\right)+\frac{12}{5} for z.

$$z=\frac{53}{32}$$

Substitute \frac{53}{32} for z in the equation y=-\frac{1}{5}z-\frac{1}{5}.

$$y=-\frac{1}{5}\times \frac{53}{32}-\frac{1}{5}$$

Calculate y from y=-\frac{1}{5}\times \frac{53}{32}-\frac{1}{5}.

$$y=-\frac{17}{32}$$

Substitute -\frac{17}{32} for y and \frac{53}{32} for z in the equation x=-2y-z+3.

$$x=-2\left(-\frac{17}{32}\right)-\frac{53}{32}+3$$

Calculate x from x=-2\left(-\frac{17}{32}\right)-\frac{53}{32}+3.

$$x=\frac{77}{32}$$

The system is now solved.

$$x=\frac{77}{32} y=-\frac{17}{32} z=\frac{53}{32}$$