Solve {l}{x+2y+3z=4}{5x+6y+7z=8}{+10y+11z=12}

$$\left. \begin{array} { l } { x + 2 y + 3 z = 4 } \\ { 5 x + 6 y + 7 z = 8 } \\ { + 10 y + 11 z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=-1$$

$$z=2$$

Short Steps Using Substitution

Solve x+2y+3z=4 for x.

$$x=-2y-3z+4$$

Substitute -2y-3z+4 for x in the equation 5x+6y+7z=8.

$$5\left(-2y-3z+4\right)+6y+7z=8$$

Solve the second equation for y and the third equation for z.

$$y=3-2z z=\frac{12}{11}-\frac{10}{11}y$$

Substitute 3-2z for y in the equation z=\frac{12}{11}-\frac{10}{11}y.

$$z=\frac{12}{11}-\frac{10}{11}\left(3-2z\right)$$

Solve z=\frac{12}{11}-\frac{10}{11}\left(3-2z\right) for z.

$$z=2$$

Substitute 2 for z in the equation y=3-2z.

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

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

$$y=-1$$

Substitute -1 for y and 2 for z in the equation x=-2y-3z+4.

$$x=-2\left(-1\right)-3\times 2+4$$

Calculate x from x=-2\left(-1\right)-3\times 2+4.

$$x=0$$

The system is now solved.

$$x=0 y=-1 z=2$$