Solve {c}{x+2y+3z=1}{5x+3y+20z=0}{3x+4y+8z=3}

$$\left\{ \begin{array} { c } { x + 2 y + 3 z = 1 } \\ { 5 x + 3 y + 20 z = 0 } \\ { 3 x + 4 y + 8 z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{37}{17} = 2\frac{3}{17} \approx 2.176470588$$

$$y=\frac{5}{17}\approx 0.294117647$$

$$z=-\frac{10}{17}\approx -0.588235294$$

Short Steps Using Substitution

Solve x+2y+3z=1 for x.

$$x=-2y-3z+1$$

Substitute -2y-3z+1 for x in the second and third equation.

$$5\left(-2y-3z+1\right)+3y+20z=0 3\left(-2y-3z+1\right)+4y+8z=3$$

Solve these equations for y and z respectively.

$$y=\frac{5}{7}+\frac{5}{7}z z=-2y$$

Substitute \frac{5}{7}+\frac{5}{7}z for y in the equation z=-2y.

$$z=-2\left(\frac{5}{7}+\frac{5}{7}z\right)$$

Solve z=-2\left(\frac{5}{7}+\frac{5}{7}z\right) for z.

$$z=-\frac{10}{17}$$

Substitute -\frac{10}{17} for z in the equation y=\frac{5}{7}+\frac{5}{7}z.

$$y=\frac{5}{7}+\frac{5}{7}\left(-\frac{10}{17}\right)$$

Calculate y from y=\frac{5}{7}+\frac{5}{7}\left(-\frac{10}{17}\right).

$$y=\frac{5}{17}$$

Substitute \frac{5}{17} for y and -\frac{10}{17} for z in the equation x=-2y-3z+1.

$$x=-2\times \frac{5}{17}-3\left(-\frac{10}{17}\right)+1$$

Calculate x from x=-2\times \frac{5}{17}-3\left(-\frac{10}{17}\right)+1.

$$x=\frac{37}{17}$$

The system is now solved.

$$x=\frac{37}{17} y=\frac{5}{17} z=-\frac{10}{17}$$