Solve the System of Linear Equations Using Elimination Method

Video Explanation

Watch the video below to understand the complete solution step by step:

Solution

Question: Solve the following system of equations:

x − y + z = 4  …… (1)

x − 2y + 3z = 9  …… (2)

2x + y + 3z = 1  …… (3)

Step 1: Eliminate x from Equations

Subtract equation (1) from equation (2):

(x − 2y + 3z) − (x − y + z) = 9 − 4

−y + 2z = 5  …… (4)

Now subtract equation (1) from equation (3):

(2x + y + 3z) − (x − y + z) = 1 − 4

x + 2y + 2z = −3  …… (5)

Step 2: Eliminate x Again

From equation (1):

x = 4 + y − z  …… (6)

Substitute x from equation (6) into equation (5):

(4 + y − z) + 2y + 2z = −3

4 + 3y + z = −3

3y + z = −7  …… (7)

Step 3: Solve Equations (4) and (7)

−y + 2z = 5  …… (4)

3y + z = −7  …… (7)

Multiply equation (7) by 2:

6y + 2z = −14  …… (8)

Subtract equation (4) from equation (8):

(6y + 2z) − (−y + 2z) = −14 − 5

7y = −19

⇒ y = −¹⁹/₇

Step 4: Find the Value of z

Substitute y = −19/7 in equation (7):

3(−19/7) + z = −7

−57/7 + z = −49/7

⇒ z = ⁸/₇

Step 5: Find the Value of x

Substitute y and z in equation (6):

x = 4 + (−19/7) − (8/7)

x = ^{28 − 27}/₇

⇒ x = ¹/₇

Final Answer

∴ The solution of the given system of equations is:

x = ¹/₇, y = −¹⁹/₇, z = ⁸/₇

Conclusion

Thus, by using the elimination method, we find that the solution of the given system of linear equations in three variables is (1/7, −19/7, 8/7).