Solve {l}{2x+y+2z=1}{y-2z=4}{3x+y=9}

$$\left\{ \begin{array} { l } { 2 x + y + 2 z = 1 } \\ { y - 2 z = 4 } \\ { 3 x + y = 9 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{13}{4} = 3\frac{1}{4} = 3.25$$

$$y=-\frac{3}{4}=-0.75$$

$$z = -\frac{19}{8} = -2\frac{3}{8} = -2.375$$

Short Steps Using Substitution

Solve 2x+y+2z=1 for y.

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

Substitute -2x-2z+1 for y in the second and third equation.

$$-2x-2z+1-2z=4 3x-2x-2z+1=9$$

Solve these equations for x and z respectively.

$$x=-2z-\frac{3}{2} z=\frac{1}{2}x-4$$

Substitute -2z-\frac{3}{2} for x in the equation z=\frac{1}{2}x-4.

$$z=\frac{1}{2}\left(-2z-\frac{3}{2}\right)-4$$

Solve z=\frac{1}{2}\left(-2z-\frac{3}{2}\right)-4 for z.

$$z=-\frac{19}{8}$$

Substitute -\frac{19}{8} for z in the equation x=-2z-\frac{3}{2}.

$$x=-2\left(-\frac{19}{8}\right)-\frac{3}{2}$$

Calculate x from x=-2\left(-\frac{19}{8}\right)-\frac{3}{2}.

$$x=\frac{13}{4}$$

Substitute \frac{13}{4} for x and -\frac{19}{8} for z in the equation y=-2x-2z+1.

$$y=-2\times \frac{13}{4}-2\left(-\frac{19}{8}\right)+1$$

Calculate y from y=-2\times \frac{13}{4}-2\left(-\frac{19}{8}\right)+1.

$$y=-\frac{3}{4}$$

The system is now solved.

$$x=\frac{13}{4} y=-\frac{3}{4} z=-\frac{19}{8}$$