Solve {l}{3x+y+z=10}{x+y-z=0}{5x-9y=1}

$$\left. \begin{array} { l } { 3 x + y + z = 10 } \\ { x + y - z = 0 } \\ { 5 x - 9 y = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=1$$

$$z=3$$

Short Steps Using Substitution

Solve 3x+y+z=10 for y.

$$y=-3x-z+10$$

Substitute -3x-z+10 for y in the second and third equation.

$$x-3x-z+10-z=0 5x-9\left(-3x-z+10\right)=1$$

Solve these equations for x and z respectively.

$$x=5-z z=\frac{91}{9}-\frac{32}{9}x$$

Substitute 5-z for x in the equation z=\frac{91}{9}-\frac{32}{9}x.

$$z=\frac{91}{9}-\frac{32}{9}\left(5-z\right)$$

Solve z=\frac{91}{9}-\frac{32}{9}\left(5-z\right) for z.

$$z=3$$

Substitute 3 for z in the equation x=5-z.

$$x=5-3$$

Calculate x from x=5-3.

$$x=2$$

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

$$y=-3\times 2-3+10$$

Calculate y from y=-3\times 2-3+10.

$$y=1$$

The system is now solved.

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