Solve {l}{2x+y+z=4}{-3x+2y-2z=-10}{x-2y+3z=7}

$$\left. \begin{array}{l}{ 2 x + y + z = 4 }\\{ - 3 x + 2 y - 2 z = - 10 }\\{ x - 2 y + 3 z = 7 }\end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=-1$$

$$z=1$$

Short Steps Using Substitution

Solve 2x+y+z=4 for y.

$$y=-2x-z+4$$

Substitute -2x-z+4 for y in the second and third equation.

$$-3x+2\left(-2x-z+4\right)-2z=-10 x-2\left(-2x-z+4\right)+3z=7$$

Solve these equations for x and z respectively.

$$x=\frac{18}{7}-\frac{4}{7}z z=3-x$$

Substitute \frac{18}{7}-\frac{4}{7}z for x in the equation z=3-x.

$$z=3-\left(\frac{18}{7}-\frac{4}{7}z\right)$$

Solve z=3-\left(\frac{18}{7}-\frac{4}{7}z\right) for z.

$$z=1$$

Substitute 1 for z in the equation x=\frac{18}{7}-\frac{4}{7}z.

$$x=\frac{18}{7}-\frac{4}{7}$$

Calculate x from x=\frac{18}{7}-\frac{4}{7}.

$$x=2$$

Substitute 2 for x and 1 for z in the equation y=-2x-z+4.

$$y=-2\times 2-1+4$$

Calculate y from y=-2\times 2-1+4.

$$y=-1$$

The system is now solved.

$$x=2 y=-1 z=1$$