SOLVING SYSTEM OF LINEAR EQUATIONS IN THREE VARIABLES

The following steps will be useful to solve system of linear equations in three  variables. 

Step 1 : 

Let us consider the system of three linear equations in three variables, say x, y and z. 

Select a pair of equations (first and second) and eliminate one of the variables, say z. After eliminating the variable, name the result as fourth equations. 

Step 2 : 

Select another pair of equations (first and third or second and third) and eliminate the same variable which you have eliminated in step 1. After eliminating the variable, name the result as fifth equations. 

Step 3 : 

Now solve the fourth and fifth equations for the two variables. 

Step 4 : 

Substitute the values of the two variables found in step 3 in one of the given three equations to solve for the third variable.

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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