SOLVING SYSTEM OF LINEAR EQUATIONS IN THREE VARIABLES

Example 1 :

Solve the following system of linear equations. 

x + y + z = 5

2x - y + z = 9

x - 2y + 3z = 16

Solution :

x + y + z = 5 -----(1)

2x - y + z = 9 -----(2)

x - 2y + 3z = 16 -----(3)

Fromm (1) and (2), eliminate z. 

(2) - (1) : 

x - 2y = 4 -----(4)

Fromm (2) and (3), eliminate z. 

3(2) - (3) : 

5x - y = 11 -----(5)

From (4) and (5), eliminate y. 

2(5) - (4) : 

9x = 18

Divide each side by 9.

x = 2

Substitute x  =  2 in (5) to solve for y. 

5(2) - y = 11

10 - y = 11

Subtract 10 from each side.

-y  =  1

Multiply each side by -1.

y = -1

Substitute x =  2 and y  =  -1 in (1). 

2 - 1 + z = 5

1 + z = 5

Subtract 1 from each side. 

z = 4

So, the solution is 

(x, y, z) = (2, -1, 4)

Example 2 :

Solve the following system of linear equations. 

3x - 3y - z = 5

2x + y - 2z = -1

x - 2y + 3z = 6

Solution :

3x - 3y - z = 5 -----(1)

2x + y - 2z = -1 -----(2)

x - 2y + 3z = 6 -----(3)

Fromm (1) and (2), eliminate z. 

2(1) - (2) : 

4x - 7y = 11 -----(4)

Fromm (1) and (3), eliminate z. 

3(1) + (3) : 

10x - 11y = 21 -----(5)

From (4) and (5), eliminate y. 

5(4) - 2(5) : 

-13y = 13

Divide each side by -13.

y = -1

Substitute y = -1 in (4) to solve for x. 

4x - 7(-1) = 11

4x + 7 = 11

Subtract 7 from each side.

4x = 4  

Multiply each side by 4.

x = 1

Substitute x = 1 and y = -1 in (3). 

1 - 2(-1) + 3z = 6

1 + 2 + 3z = 6

3 + 3z = 6

Subtract 3 from each side. 

3z = 3

Divide each side by 3.

z = 1

So, the solution is 

(x, y, z) = (1, -1, 1)

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