Solve {l}{6x+7y-3z=3}{9x-2y+12z=20}{3x+2y-3z=-1}

$$\left. \begin{array} { l } { 6 x + 7 y - 3 z = 3 } \\ { 9 x - 2 y + 12 z = 20 } \\ { 3 x + 2 y - 3 z = - 1 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{56}{87}\approx 0.643678161$$

$$y=\frac{12}{29}\approx 0.413793103$$

$$z = \frac{109}{87} = 1\frac{22}{87} \approx 1.252873563$$

Short Steps Using Substitution

Solve 6x+7y-3z=3 for x.

$$x=-\frac{7}{6}y+\frac{1}{2}z+\frac{1}{2}$$

Substitute -\frac{7}{6}y+\frac{1}{2}z+\frac{1}{2} for x in the second and third equation.

$$9\left(-\frac{7}{6}y+\frac{1}{2}z+\frac{1}{2}\right)-2y+12z=20 3\left(-\frac{7}{6}y+\frac{1}{2}z+\frac{1}{2}\right)+2y-3z=-1$$

Solve these equations for y and z respectively.

$$y=-\frac{31}{25}+\frac{33}{25}z z=\frac{5}{3}-y$$

Substitute -\frac{31}{25}+\frac{33}{25}z for y in the equation z=\frac{5}{3}-y.

$$z=\frac{5}{3}-\left(-\frac{31}{25}+\frac{33}{25}z\right)$$

Solve z=\frac{5}{3}-\left(-\frac{31}{25}+\frac{33}{25}z\right) for z.

$$z=\frac{109}{87}$$

Substitute \frac{109}{87} for z in the equation y=-\frac{31}{25}+\frac{33}{25}z.

$$y=-\frac{31}{25}+\frac{33}{25}\times \frac{109}{87}$$

Calculate y from y=-\frac{31}{25}+\frac{33}{25}\times \frac{109}{87}.

$$y=\frac{12}{29}$$

Substitute \frac{12}{29} for y and \frac{109}{87} for z in the equation x=-\frac{7}{6}y+\frac{1}{2}z+\frac{1}{2}.

$$x=-\frac{7}{6}\times \frac{12}{29}+\frac{1}{2}\times \frac{109}{87}+\frac{1}{2}$$

Calculate x from x=-\frac{7}{6}\times \frac{12}{29}+\frac{1}{2}\times \frac{109}{87}+\frac{1}{2}.

$$x=\frac{56}{87}$$

The system is now solved.

$$x=\frac{56}{87} y=\frac{12}{29} z=\frac{109}{87}$$