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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