Solve {l}{6x+2y-3z=12}{2x+6y-z=11}{15x+2y-6z=45}

$$\left. \begin{array} { l } { 6 x + 2 y - 3 z = 12 } \\ { 2 x + 6 y - z = 11 } \\ { 15 x + 2 y - 6 z = 45 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{63}{8} = 7\frac{7}{8} = 7.875$$

$$y = \frac{21}{16} = 1\frac{5}{16} = 1.3125$$

$$z = \frac{101}{8} = 12\frac{5}{8} = 12.625$$

Short Steps Using Substitution

Reorder the equations.

$$2x+6y-z=11 6x+2y-3z=12 15x+2y-6z=45$$

Solve 2x+6y-z=11 for z.

$$z=2x+6y-11$$

Substitute 2x+6y-11 for z in the second and third equation.

$$6x+2y-3\left(2x+6y-11\right)=12 15x+2y-6\left(2x+6y-11\right)=45$$

Solve these equations for y and x respectively.

$$y=\frac{21}{16} x=\frac{34}{3}y-7$$

Substitute \frac{21}{16} for y in the equation x=\frac{34}{3}y-7.

$$x=\frac{34}{3}\times \frac{21}{16}-7$$

Calculate x from x=\frac{34}{3}\times \frac{21}{16}-7.

$$x=\frac{63}{8}$$

Substitute \frac{21}{16} for y and \frac{63}{8} for x in the equation z=2x+6y-11.

$$z=2\times \frac{63}{8}+6\times \frac{21}{16}-11$$

Calculate z from z=2\times \frac{63}{8}+6\times \frac{21}{16}-11.

$$z=\frac{101}{8}$$

The system is now solved.

$$x=\frac{63}{8} y=\frac{21}{16} z=\frac{101}{8}$$