Solve {l}{6x-7y=9}{y+2z=2}{z+5x=1}

$$\left. \begin{array} { l } { 6 x - 7 y = 9 } \\ { y + 2 z = 2 } \\ { z + 5 x = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{9}{64}=-0.140625$$

$$y = -\frac{45}{32} = -1\frac{13}{32} = -1.40625$$

$$z = \frac{109}{64} = 1\frac{45}{64} = 1.703125$$

Short Steps Using Substitution

Reorder the equations.

$$y+2z=2 6x-7y=9 z+5x=1$$

Solve y+2z=2 for y.

$$y=2-2z$$

Substitute 2-2z for y in the equation 6x-7y=9.

$$6x-7\left(2-2z\right)=9$$

Solve the second equation for x and the third equation for z.

$$x=\frac{23}{6}-\frac{7}{3}z z=-5x+1$$

Substitute \frac{23}{6}-\frac{7}{3}z for x in the equation z=-5x+1.

$$z=-5\left(\frac{23}{6}-\frac{7}{3}z\right)+1$$

Solve z=-5\left(\frac{23}{6}-\frac{7}{3}z\right)+1 for z.

$$z=\frac{109}{64}$$

Substitute \frac{109}{64} for z in the equation x=\frac{23}{6}-\frac{7}{3}z.

$$x=\frac{23}{6}-\frac{7}{3}\times \frac{109}{64}$$

Calculate x from x=\frac{23}{6}-\frac{7}{3}\times \frac{109}{64}.

$$x=-\frac{9}{64}$$

Substitute \frac{109}{64} for z in the equation y=2-2z.

$$y=2-2\times \frac{109}{64}$$

Calculate y from y=2-2\times \frac{109}{64}.

$$y=-\frac{45}{32}$$

The system is now solved.

$$x=-\frac{9}{64} y=-\frac{45}{32} z=\frac{109}{64}$$