Solve {l}{5x+y+z=24}{x+5y+z=-4}{x+y+5z=36}

$$\left\{ \begin{array} { l } { 5 x + y + z = 24 } \\ { x + 5 y + z = - 4 } \\ { x + y + 5 z = 36 } \end{array} \right.$$

Solve for x, y, z

$$x=4$$

$$y=-3$$

$$z=7$$

Short Steps Using Substitution

Solve 5x+y+z=24 for y.

$$y=-5x-z+24$$

Substitute -5x-z+24 for y in the second and third equation.

$$x+5\left(-5x-z+24\right)+z=-4 x-5x-z+24+5z=36$$

Solve these equations for x and z respectively.

$$x=\frac{31}{6}-\frac{1}{6}z z=3+x$$

Substitute \frac{31}{6}-\frac{1}{6}z for x in the equation z=3+x.

$$z=3+\frac{31}{6}-\frac{1}{6}z$$

Solve z=3+\frac{31}{6}-\frac{1}{6}z for z.

$$z=7$$

Substitute 7 for z in the equation x=\frac{31}{6}-\frac{1}{6}z.

$$x=\frac{31}{6}-\frac{1}{6}\times 7$$

Calculate x from x=\frac{31}{6}-\frac{1}{6}\times 7.

$$x=4$$

Substitute 4 for x and 7 for z in the equation y=-5x-z+24.

$$y=-5\times 4-7+24$$

Calculate y from y=-5\times 4-7+24.

$$y=-3$$

The system is now solved.

$$x=4 y=-3 z=7$$