Solve {l}{3x+5y+17z=63}{5x+6y+21z=79}{x+11y+23z=42}

$$\left\{ \begin{array} { l } { 3 x + 5 y + 17 z = 63 } \\ { 5 x + 6 y + 21 z = 79 } \\ { x + 11 y + 23 z = 42 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{15}{28}\approx -0.535714286$$

$$y = -\frac{445}{42} = -10\frac{25}{42} \approx -10.595238095$$

$$z = \frac{83}{12} = 6\frac{11}{12} \approx 6.916666667$$

Short Steps Using Substitution

Reorder the equations.

$$x+11y+23z=42 5x+6y+21z=79 3x+5y+17z=63$$

Solve x+11y+23z=42 for x.

$$x=-11y-23z+42$$

Substitute -11y-23z+42 for x in the second and third equation.

$$5\left(-11y-23z+42\right)+6y+21z=79 3\left(-11y-23z+42\right)+5y+17z=63$$

Solve these equations for y and z respectively.

$$y=-\frac{94}{49}z+\frac{131}{49} z=\frac{63}{52}-\frac{7}{13}y$$

Substitute -\frac{94}{49}z+\frac{131}{49} for y in the equation z=\frac{63}{52}-\frac{7}{13}y.

$$z=\frac{63}{52}-\frac{7}{13}\left(-\frac{94}{49}z+\frac{131}{49}\right)$$

Solve z=\frac{63}{52}-\frac{7}{13}\left(-\frac{94}{49}z+\frac{131}{49}\right) for z.

$$z=\frac{83}{12}$$

Substitute \frac{83}{12} for z in the equation y=-\frac{94}{49}z+\frac{131}{49}.

$$y=-\frac{94}{49}\times \frac{83}{12}+\frac{131}{49}$$

Calculate y from y=-\frac{94}{49}\times \frac{83}{12}+\frac{131}{49}.

$$y=-\frac{445}{42}$$

Substitute -\frac{445}{42} for y and \frac{83}{12} for z in the equation x=-11y-23z+42.

$$x=-11\left(-\frac{445}{42}\right)-23\times \frac{83}{12}+42$$

Calculate x from x=-11\left(-\frac{445}{42}\right)-23\times \frac{83}{12}+42.

$$x=-\frac{15}{28}$$

The system is now solved.

$$x=-\frac{15}{28} y=-\frac{445}{42} z=\frac{83}{12}$$