Solve {l}{x+y-z+2u=10}{3x-y+7z+4y=1}{-5x+3y-15z-6y=9}

$$\left. \begin{array}{l}{ x + y - z + 2 u = 10 }\\{ 3 x - y + 7 z + 4 y = 1 }\\{ - 5 x + 3 y - 15 z - 6 y = 9 }\end{array} \right.$$

Solve for x, y, z

$$x=\frac{33-12u}{5}$$

$$y = \frac{1}{2} = 0.5$$

$$z=\frac{3u}{5}-\frac{29}{10}$$

Short Steps Using Substitution

Solve x+y-z+2u=10 for x.

$$x=-y+z-2u+10$$

Substitute -y+z-2u+10 for x in the second and third equation.

$$3\left(-y+z-2u+10\right)-y+7z+4y=1 -5\left(-y+z-2u+10\right)+3y-15z-6y=9$$

Solve these equations for z and y respectively.

$$z=-\frac{29}{10}+\frac{3}{5}u y=10z-5u+\frac{59}{2}$$

Substitute -\frac{29}{10}+\frac{3}{5}u for z in the equation y=10z-5u+\frac{59}{2}.

$$y=10\left(-\frac{29}{10}+\frac{3}{5}u\right)-5u+\frac{59}{2}$$

Calculate y from y=10\left(-\frac{29}{10}+\frac{3}{5}u\right)-5u+\frac{59}{2}.

$$y=\frac{1}{2}+u$$

Substitute -\frac{29}{10}+\frac{3}{5}u for z and \frac{1}{2}+u for y in the equation x=-y+z-2u+10.

$$x=-\left(\frac{1}{2}+u\right)-\frac{29}{10}+\frac{3}{5}u-2u+10$$

Calculate x from x=-\left(\frac{1}{2}+u\right)-\frac{29}{10}+\frac{3}{5}u-2u+10.

$$x=\frac{33}{5}-\frac{12}{5}u$$

The system is now solved.

$$x=\frac{33}{5}-\frac{12}{5}u y=\frac{1}{2}+u z=-\frac{29}{10}+\frac{3}{5}u$$