Solve {l}{15x-2y+z=19}{2*3x+y+z=4}{3*3x+y-z=12}

$$\left. \begin{array} { l } { 15 x - 2 y + z = 19 } \\ { 2 \cdot 3 x + y + z = 4 } \\ { 3 \cdot 3 x + y - z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{26}{21} = 1\frac{5}{21} \approx 1.238095238$$

$$y = -\frac{9}{7} = -1\frac{2}{7} \approx -1.285714286$$

$$z = -\frac{15}{7} = -2\frac{1}{7} \approx -2.142857143$$

Short Steps Using Substitution

Solve 15x-2y+z=19 for z.

$$z=-15x+2y+19$$

Substitute -15x+2y+19 for z in the second and third equation.

$$2\times 3x+y-15x+2y+19=4 3\times 3x+y-\left(-15x+2y+19\right)=12$$

Solve these equations for y and x respectively.

$$y=-5+3x x=\frac{31}{24}+\frac{1}{24}y$$

Substitute -5+3x for y in the equation x=\frac{31}{24}+\frac{1}{24}y.

$$x=\frac{31}{24}+\frac{1}{24}\left(-5+3x\right)$$

Solve x=\frac{31}{24}+\frac{1}{24}\left(-5+3x\right) for x.

$$x=\frac{26}{21}$$

Substitute \frac{26}{21} for x in the equation y=-5+3x.

$$y=-5+3\times \frac{26}{21}$$

Calculate y from y=-5+3\times \frac{26}{21}.

$$y=-\frac{9}{7}$$

Substitute -\frac{9}{7} for y and \frac{26}{21} for x in the equation z=-15x+2y+19.

$$z=-15\times \frac{26}{21}+2\left(-\frac{9}{7}\right)+19$$

Calculate z from z=-15\times \frac{26}{21}+2\left(-\frac{9}{7}\right)+19.

$$z=-\frac{15}{7}$$

The system is now solved.

$$x=\frac{26}{21} y=-\frac{9}{7} z=-\frac{15}{7}$$