Solve {l}{5x+6y+4z=91}{x+y+z=51}{3x+4y+6z=180}

$$\left. \begin{array} { l } { 5 x + 6 y + 4 z = 91 } \\ { x + y + z = 51 } \\ { 3 x + 4 y + 6 z = 180 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{239}{2} = 119\frac{1}{2} = 119.5$$

$$y = -\frac{465}{4} = -116\frac{1}{4} = -116.25$$

$$z = \frac{191}{4} = 47\frac{3}{4} = 47.75$$

Short Steps Using Substitution

Reorder the equations.

$$x+y+z=51 5x+6y+4z=91 3x+4y+6z=180$$

Solve x+y+z=51 for x.

$$x=-y-z+51$$

Substitute -y-z+51 for x in the second and third equation.

$$5\left(-y-z+51\right)+6y+4z=91 3\left(-y-z+51\right)+4y+6z=180$$

Solve these equations for y and z respectively.

$$y=z-164 z=-\frac{1}{3}y+9$$

Substitute z-164 for y in the equation z=-\frac{1}{3}y+9.

$$z=-\frac{1}{3}\left(z-164\right)+9$$

Solve z=-\frac{1}{3}\left(z-164\right)+9 for z.

$$z=\frac{191}{4}$$

Substitute \frac{191}{4} for z in the equation y=z-164.

$$y=\frac{191}{4}-164$$

Calculate y from y=\frac{191}{4}-164.

$$y=-\frac{465}{4}$$

Substitute -\frac{465}{4} for y and \frac{191}{4} for z in the equation x=-y-z+51.

$$x=-\left(-\frac{465}{4}\right)-\frac{191}{4}+51$$

Calculate x from x=-\left(-\frac{465}{4}\right)-\frac{191}{4}+51.

$$x=\frac{239}{2}$$

The system is now solved.

$$x=\frac{239}{2} y=-\frac{465}{4} z=\frac{191}{4}$$