Solve {l}{3x+5y-z=4}{10y-6x-3z=1}{4z-15y+9x=-1}

$$\left\{ \begin{array} { l } { 3 x + 5 y - z = 4 } \\ { 10 y - 6 x - 3 z = 1 } \\ { 4 z - 15 y + 9 x = - 1 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{2}{3}\approx 0.666666667$$

$$y=\frac{1}{5}=0.2$$

$$z=-1$$

Short Steps Using Substitution

Solve 3x+5y-z=4 for z.

$$z=3x+5y-4$$

Substitute 3x+5y-4 for z in the second and third equation.

$$10y-6x-3\left(3x+5y-4\right)=1 4\left(3x+5y-4\right)-15y+9x=-1$$

Solve these equations for y and x respectively.

$$y=-3x+\frac{11}{5} x=\frac{5}{7}-\frac{5}{21}y$$

Substitute -3x+\frac{11}{5} for y in the equation x=\frac{5}{7}-\frac{5}{21}y.

$$x=\frac{5}{7}-\frac{5}{21}\left(-3x+\frac{11}{5}\right)$$

Solve x=\frac{5}{7}-\frac{5}{21}\left(-3x+\frac{11}{5}\right) for x.

$$x=\frac{2}{3}$$

Substitute \frac{2}{3} for x in the equation y=-3x+\frac{11}{5}.

$$y=-3\times \frac{2}{3}+\frac{11}{5}$$

Calculate y from y=-3\times \frac{2}{3}+\frac{11}{5}.

$$y=\frac{1}{5}$$

Substitute \frac{1}{5} for y and \frac{2}{3} for x in the equation z=3x+5y-4.

$$z=3\times \frac{2}{3}+5\times \frac{1}{5}-4$$

Calculate z from z=3\times \frac{2}{3}+5\times \frac{1}{5}-4.

$$z=-1$$

The system is now solved.

$$x=\frac{2}{3} y=\frac{1}{5} z=-1$$