Solve {l}{5x+10y+14z=-1}{8x+15y+22z=4}{x+2y+3z=-4}

$$\left\{ \begin{array}{l}{ 5 x + 10 y + 14 z = - 1 }\\{ 8 x + 15 y + 22 z = 4 }\\{ x + 2 y + 3 z = - 4 }\end{array} \right.$$

Solve for x, y, z

$$x=49$$

$$y=2$$

$$z=-19$$

Short Steps Using Substitution

Reorder the equations.

$$x+2y+3z=-4 8x+15y+22z=4 5x+10y+14z=-1$$

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

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

Substitute -2y-3z-4 for x in the second and third equation.

$$8\left(-2y-3z-4\right)+15y+22z=4 5\left(-2y-3z-4\right)+10y+14z=-1$$

Solve these equations for y and z respectively.

$$y=-2z-36 z=-19$$

Substitute -19 for z in the equation y=-2z-36.

$$y=-2\left(-19\right)-36$$

Calculate y from y=-2\left(-19\right)-36.

$$y=2$$

Substitute 2 for y and -19 for z in the equation x=-2y-3z-4.

$$x=-2\times 2-3\left(-19\right)-4$$

Calculate x from x=-2\times 2-3\left(-19\right)-4.

$$x=49$$

The system is now solved.

$$x=49 y=2 z=-19$$