Solve {l}{3X+4Y+2Z=19}{6X+Y-Z=7}{-5X-2Y+Z=-9}

$$\left. \begin{array} { l } { 3 X + 4 Y + 2 Z = 19 } \\ { 6 X + Y - Z = 7 } \\ { - 5 X - 2 Y + Z = - 9 } \end{array} \right.$$

Solve for X, Y, Z

$$X=1$$

$$Y=3$$

$$Z=2$$

Short Steps Using Substitution

Reorder the equations.

$$6X+Y-Z=7 3X+4Y+2Z=19 -5X-2Y+Z=-9$$

Solve 6X+Y-Z=7 for Y.

$$Y=-6X+Z+7$$

Substitute -6X+Z+7 for Y in the second and third equation.

$$3X+4\left(-6X+Z+7\right)+2Z=19 -5X-2\left(-6X+Z+7\right)+Z=-9$$

Solve these equations for X and Z respectively.

$$X=\frac{3}{7}+\frac{2}{7}Z Z=7X-5$$

Substitute \frac{3}{7}+\frac{2}{7}Z for X in the equation Z=7X-5.

$$Z=7\left(\frac{3}{7}+\frac{2}{7}Z\right)-5$$

Solve Z=7\left(\frac{3}{7}+\frac{2}{7}Z\right)-5 for Z.

$$Z=2$$

Substitute 2 for Z in the equation X=\frac{3}{7}+\frac{2}{7}Z.

$$X=\frac{3}{7}+\frac{2}{7}\times 2$$

Calculate X from X=\frac{3}{7}+\frac{2}{7}\times 2.

$$X=1$$

Substitute 1 for X and 2 for Z in the equation Y=-6X+Z+7.

$$Y=-6+2+7$$

Calculate Y from Y=-6+2+7.

$$Y=3$$

The system is now solved.

$$X=1 Y=3 Z=2$$