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

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

Solve for x, y, z

$$x=2$$

$$y=1$$

$$z=-4$$

Short Steps Using Substitution

Solve 3x+2y+z=4 for z.

$$z=-3x-2y+4$$

Substitute -3x-2y+4 for z in the second and third equation.

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

Solve these equations for y and x respectively.

$$y=-\frac{4}{7}x+\frac{15}{7} x=\frac{7}{2}-\frac{3}{2}y$$

Substitute -\frac{4}{7}x+\frac{15}{7} for y in the equation x=\frac{7}{2}-\frac{3}{2}y.

$$x=\frac{7}{2}-\frac{3}{2}\left(-\frac{4}{7}x+\frac{15}{7}\right)$$

Solve x=\frac{7}{2}-\frac{3}{2}\left(-\frac{4}{7}x+\frac{15}{7}\right) for x.

$$x=2$$

Substitute 2 for x in the equation y=-\frac{4}{7}x+\frac{15}{7}.

$$y=-\frac{4}{7}\times 2+\frac{15}{7}$$

Calculate y from y=-\frac{4}{7}\times 2+\frac{15}{7}.

$$y=1$$

Substitute 1 for y and 2 for x in the equation z=-3x-2y+4.

$$z=-3\times 2-2+4$$

Calculate z from z=-3\times 2-2+4.

$$z=-4$$

The system is now solved.

$$x=2 y=1 z=-4$$