SOLVING SYSTEM OF LINEAR EQUATIONS IN THREE VARIABLES

Solving System of Linear Equations in Three Variables :

Here we are going to see, how to solve system of linear equations in three variables.

Procedure for solving system of linear equations in three variables :

Step 1 :

By taking any two equations from the given three, first multiply by some suitable non-zero constant to make the co-efficient of one variable (either x or y or z) numerically equal.

Step 2 :

Eliminate one of the variables whose co-efficients are numerically equal from the equations.

Step 3 :

Eliminate the same variable from another pair.

Step 4 :

Now we have two equations in two variables.

Step 5 :

Solve them using any method studied in earlier classes

Step 6 :

The remaining variable is then found by substituting in any one of the given equations.

Solving Systems of Linear Equations in Three Variables Example Problems

Question 1 :

Solve the following system of linear equations in three variables

(i) 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)

Let us add (1) and (2)

Multiply the first equation by 2 and add by (3)

Let 3x + 2z  =  14  ----(4) and 3x + 5z = 26  ---(5)

(4) - (5) 

  3x + 2z  =  14

  3x + 5z  =  26

  (-)   (-)      (-)

  ----------------

      -3z  =  -12  ==> z  =  4

By applying z = 4 in (4), we get

3x + 2(4)  =  14

3x + 8  =  12

3x  =  14 - 8

3x  =  6  ==>  x  =  2

By applying x 2 and z = 4 in (1), we get 

2 + y + 4  =  5

6 + y  =  5 

y  =  5 - 6

y  =  -1

Hence the required solution is x = 2, y = -1 and z = 2.


After having gone through the stuff given above, we hope that the students would have understood, "Solving System of Linear Equations in Three Variables". 

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