# Question #9ab83

Mar 23, 2017

$x = 4 , y = 0 , z = - 3$

#### Explanation:

Given three equation and three unknown
$x + y - z = 7$ .......(1)
$- 2 x + 2 y - z = - 5$ .......(2)
$3 x - 3 y + 2 z = 6$ ......(3)

Step 1. Eliminate one unknown from first two equations. Let us take $z$

Subtracting (2) from (1) we get
$\left(x + y - z\right) - \left(- 2 x + 2 y - z\right) = 7 - \left(- 5\right)$
$\implies x + y - z + 2 x - 2 y + z = 7 + 5$
$\implies 3 x - y = 12$ ......(4)

Step 2. Eliminate same unknown from next two equations. (one can choose last two as well).

Multiplying (2) with $2$ and adding (3)
$2 \times \left(- 2 x + 2 y - z\right) + \left(3 x - 3 y + 2 z\right) = 2 \times \left(- 5\right) + 6$
$\implies - 4 x + 4 y - 2 z + 3 x - 3 y + 2 z = - 10 + 6$
$\implies - x + y = - 4$ .....(5)

We now have two equations (4) and (5) with two unknowns.
Step 3. Eliminate one unknown from these two equations.

We see that if we add these two we eliminate $y$. We get
$3 x - y + \left(- x + y\right) = 12 + \left(- 4\right)$
$\implies 3 x - y - x + y = 12 - 4$
$\implies 2 x = 8$
$\implies x = \frac{8}{2}$
$\implies x = 4$

Step 4. Insert this value of $x$ in either (4) or (5) to obtain $y$. Let us choose (5)

$- 4 + y = - 4$
$y = - 4 + 4$
$y = 0$

Step 5. Insert these values of $x \mathmr{and} y$ in any of (1), (2) or (3) to obtain $z$. Let us choose (1)

$4 + 0 - z = 7$
$\implies z = 4 - 7$
$\implies z = - 3$