Solve {l}{2x+3y-2z=3}{4x+y-5z=0}{x+4y+2z=12}

$$\left\{ \begin{array} { l } { 2 x + 3 y - 2 z = 3 } \\ { 4 x + y - 5 z = 0 } \\ { x + 4 y + 2 z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{18}{5} = 3\frac{3}{5} = 3.6$$

$$y=\frac{3}{5}=0.6$$

$$z=3$$

Short Steps Using Substitution

Reorder the equations.

$$4x+y-5z=0 2x+3y-2z=3 x+4y+2z=12$$

Solve 4x+y-5z=0 for y.

$$y=-4x+5z$$

Substitute -4x+5z for y in the second and third equation.

$$2x+3\left(-4x+5z\right)-2z=3 x+4\left(-4x+5z\right)+2z=12$$

Solve these equations for x and z respectively.

$$x=\frac{13}{10}z-\frac{3}{10} z=\frac{6}{11}+\frac{15}{22}x$$

Substitute \frac{13}{10}z-\frac{3}{10} for x in the equation z=\frac{6}{11}+\frac{15}{22}x.

$$z=\frac{6}{11}+\frac{15}{22}\left(\frac{13}{10}z-\frac{3}{10}\right)$$

Solve z=\frac{6}{11}+\frac{15}{22}\left(\frac{13}{10}z-\frac{3}{10}\right) for z.

$$z=3$$

Substitute 3 for z in the equation x=\frac{13}{10}z-\frac{3}{10}.

$$x=\frac{13}{10}\times 3-\frac{3}{10}$$

Calculate x from x=\frac{13}{10}\times 3-\frac{3}{10}.

$$x=\frac{18}{5}$$

Substitute \frac{18}{5} for x and 3 for z in the equation y=-4x+5z.

$$y=-4\times \frac{18}{5}+5\times 3$$

Calculate y from y=-4\times \frac{18}{5}+5\times 3.

$$y=\frac{3}{5}$$

The system is now solved.

$$x=\frac{18}{5} y=\frac{3}{5} z=3$$