Solve {l}{2x+y-2z=10}{x+y+4z=-9}{7x+5y+z=14}

$$\left\{ \begin{array}{l}{ 2 x + y - 2 z = 10 }\\{ x + y + 4 z = - 9 }\\{ 7 x + 5 y + z = 14 }\end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=-3$$

Short Steps Using Substitution

Solve 2x+y-2z=10 for y.

$$y=10-2x+2z$$

Substitute 10-2x+2z for y in the second and third equation.

$$x+10-2x+2z+4z=-9 7x+5\left(10-2x+2z\right)+z=14$$

Solve these equations for x and z respectively.

$$x=19+6z z=-\frac{36}{11}+\frac{3}{11}x$$

Substitute 19+6z for x in the equation z=-\frac{36}{11}+\frac{3}{11}x.

$$z=-\frac{36}{11}+\frac{3}{11}\left(19+6z\right)$$

Solve z=-\frac{36}{11}+\frac{3}{11}\left(19+6z\right) for z.

$$z=-3$$

Substitute -3 for z in the equation x=19+6z.

$$x=19+6\left(-3\right)$$

Calculate x from x=19+6\left(-3\right).

$$x=1$$

Substitute 1 for x and -3 for z in the equation y=10-2x+2z.

$$y=10-2+2\left(-3\right)$$

Calculate y from y=10-2+2\left(-3\right).

$$y=2$$

The system is now solved.

$$x=1 y=2 z=-3$$