Examples — lesson. Mathematics State Board, Class 10.

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Let us learn how to find the solution to the system of linear equations 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

$2$ .

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

$-6x + 6y + 3z = 9$

__________________________

$10y + z = 21$ ---- (

$5$ )

Step 3: Solve equations (

$4$ ) and (

$5$ ).

$4y + 3z = 11$

$30y + 3z = 63$

(

$-$ ) (

$-$ )

___________________________

$- 26y = - 52$

$y = 2$

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

$z = 1$ .

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5. Examples

Theory: