# How do you solve the system 3x - 4y - 2z = 13, 2x + y + 2z = 5, 6x +2y - z = 16?

Jan 26, 2018

$\textcolor{b l u e}{x = 3 , y = - 1 , z = 0}$

#### Explanation:

$3 x - 4 y - 2 z = 13$ Eqn (1)

$2 x + y + 2 z = 5$ Eqn (2)

$6 x + 2 y - z = 16$ Eqn (3)

Add Eqn (1), (2) to eliminate z

$5 x - 3 y = 18$ Eqn (4)

Multiply Eqn (3) by 2 and add to Eqn (2)

$14 x + 5 y = 37$ Eqn (5)

Find x & y by solving equations (4) & (5)

Multiply Eqn (4) by 5

$25 x - 15 y = 90$ Eqn (6)

Multiply Eqn (5) by 3

$42 x + 15 y = 111$ Eqn (7)

$67 x = 201$ or $x = \textcolor{p u r p \le}{3}$

Substituting value of x in Eqn (4)

$\left(5 \cdot 3\right) - 3 y = 18$

$3 y = 15 - 18 = - 3$, $y = \textcolor{p u r p \le}{- 1}$

Substituting values of x & y in Eqn (3)

$\left(6 \cdot 3\right) + \left(2 \cdot \left(- 1\right)\right) - z = 16$

$z = 18 - 2 - 16 = 0$

$z = \textcolor{p u r p \le}{0}$

$\textcolor{b l u e}{x = 3 , y = - 1 , z = - 0}$