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### Theory:

Let us learn how to find the solution to the system of linear equation in three variables.
Example:
Solve the system of linear equations in three variables $$6x + 4y - 2z = 12$$, $$-2x + 2y + z = 3$$, $$2x + 2y + 2z = 8$$.

Solution:

Let us name the equations.

$$6x + 4y - 2z = 12$$ ---- ($$1$$)

$$-2x + 2y + z = 3$$ ---- ($$2$$)

$$2x + 2y + 2z = 8$$ ---- ($$3$$)

Step 1: Solving equations ($$2$$) and ($$3$$).

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

$$2x + 2y + 2z = 8$$
__________________________
$$4y + 3z = 11$$ ---- ($$4$$)

Step 2: Similarly, let us eliminate the variable $$x$$ from equations $$1$$ and $$2$$.

$$6x + 4y -2z = 12$$

$$-6x + 6y + 3z = 9$$
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$$10y + z = 21$$ ---- ($$5$$)

Step 3: Solve equations ($$4$$) and ($$5$$).

$$4y + 3z = 11$$

$$30y + 3z = 63$$
($$-$$)    ($$-$$)      ($$-$$)
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$$- 26y = - 52$$

$$y = 2$$

Substitute the value of $$y$$ in equation ($$5$$), we get:

$$20 + z = 21$$

$$z = 1$$

Step 4: Substitute the value of $$y$$ and $$z$$ in equation ($$1$$), we get:

$$6x + 4(2) - 2(1) = 12$$

$$6x + 8 - 2 = 12$$

$$6x + 6 = 12$$

$$6x = 6$$

$$x = 1$$

Therefore, the solution is $$x = 1$$, $$y = 2$$ and $$z = 1$$.