Solve {l}{2x+4y+5z=23}{3x-y-2z=6}{4x+2y+5z=31}

$$\left. \begin{array} { l } { 2 x + 4 y + 5 z = 23 } \\ { 3 x - y - 2 z = 6 } \\ { 4 x + 2 y + 5 z = 31 } \end{array} \right.$$

Solve for x, y, z

$$x=4$$

$$y=0$$

$$z=3$$

Short Steps Using Substitution

Reorder the equations.

$$3x-y-2z=6 2x+4y+5z=23 4x+2y+5z=31$$

Solve 3x-y-2z=6 for y.

$$y=3x-2z-6$$

Substitute 3x-2z-6 for y in the second and third equation.

$$2x+4\left(3x-2z-6\right)+5z=23 4x+2\left(3x-2z-6\right)+5z=31$$

Solve these equations for x and z respectively.

$$x=\frac{3}{14}z+\frac{47}{14} z=-10x+43$$

Substitute \frac{3}{14}z+\frac{47}{14} for x in the equation z=-10x+43.

$$z=-10\left(\frac{3}{14}z+\frac{47}{14}\right)+43$$

Solve z=-10\left(\frac{3}{14}z+\frac{47}{14}\right)+43 for z.

$$z=3$$

Substitute 3 for z in the equation x=\frac{3}{14}z+\frac{47}{14}.

$$x=\frac{3}{14}\times 3+\frac{47}{14}$$

Calculate x from x=\frac{3}{14}\times 3+\frac{47}{14}.

$$x=4$$

Substitute 4 for x and 3 for z in the equation y=3x-2z-6.

$$y=3\times 4-2\times 3-6$$

Calculate y from y=3\times 4-2\times 3-6.

$$y=0$$

The system is now solved.

$$x=4 y=0 z=3$$