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

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

Solve for x, y, z

$$x=1$$

$$y=-1$$

$$z=2$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for x and z respectively.

$$x=-\frac{4}{5}z+\frac{13}{5} z=4-2x$$

Substitute -\frac{4}{5}z+\frac{13}{5} for x in the equation z=4-2x.

$$z=4-2\left(-\frac{4}{5}z+\frac{13}{5}\right)$$

Solve z=4-2\left(-\frac{4}{5}z+\frac{13}{5}\right) for z.

$$z=2$$

Substitute 2 for z in the equation x=-\frac{4}{5}z+\frac{13}{5}.

$$x=-\frac{4}{5}\times 2+\frac{13}{5}$$

Calculate x from x=-\frac{4}{5}\times 2+\frac{13}{5}.

$$x=1$$

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

$$y=-2-2+3$$

Calculate y from y=-2-2+3.

$$y=-1$$

The system is now solved.

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