Solve {l}{x+y+z=3}{3x+2y-z=10}{2x-5y-z=-5}

$$\left. \begin{array} { l } { x + y + z = 3 } \\ { 3 x + 2 y - z = 10 } \\ { 2 x - 5 y - z = - 5 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{46}{25} = 1\frac{21}{25} = 1.84$$

$$y = \frac{47}{25} = 1\frac{22}{25} = 1.88$$

$$z=-\frac{18}{25}=-0.72$$

Short Steps Using Substitution

Solve x+y+z=3 for x.

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

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

$$3\left(-y-z+3\right)+2y-z=10 2\left(-y-z+3\right)-5y-z=-5$$

Solve these equations for y and z respectively.

$$y=-4z-1 z=-\frac{7}{3}y+\frac{11}{3}$$

Substitute -4z-1 for y in the equation z=-\frac{7}{3}y+\frac{11}{3}.

$$z=-\frac{7}{3}\left(-4z-1\right)+\frac{11}{3}$$

Solve z=-\frac{7}{3}\left(-4z-1\right)+\frac{11}{3} for z.

$$z=-\frac{18}{25}$$

Substitute -\frac{18}{25} for z in the equation y=-4z-1.

$$y=-4\left(-\frac{18}{25}\right)-1$$

Calculate y from y=-4\left(-\frac{18}{25}\right)-1.

$$y=\frac{47}{25}$$

Substitute \frac{47}{25} for y and -\frac{18}{25} for z in the equation x=-y-z+3.

$$x=-\frac{47}{25}-\left(-\frac{18}{25}\right)+3$$

Calculate x from x=-\frac{47}{25}-\left(-\frac{18}{25}\right)+3.

$$x=\frac{46}{25}$$

The system is now solved.

$$x=\frac{46}{25} y=\frac{47}{25} z=-\frac{18}{25}$$