Q.40 The solution of the following set of equations is
x + 2y + 3z = 20
7x + 3y + z = 13
x + 6y + 2z = 0
- x = −2, y = 2, z = 8
- x = 2, y = −3, z = 8
- x = 2, y = 3, z = −8
- x = 8, y = 2, z = −3
Solving a System of Linear Equations with Three Variables
Solving a system of linear equations involving three variables is a common problem in competitive examinations.
In this solution, we will solve the given system step by step and analyze all the answer options.
Step-by-Step Solution
Step 1: Eliminate x
Subtract equation (1) from equation (3):
(x + 6y + 2z) − (x + 2y + 3z) = 0 − 20
4y − z = −20 …(A)
Subtract 7 × equation (1) from equation (2):
(7x + 3y + z) − 7(x + 2y + 3z) = 13 − 140
−11y − 20z = −127 …(B)
Step 2: Solve Equations (A) and (B)
From equation (A):
z = 4y + 20
Substitute z into equation (B):
−11y − 20(4y + 20) = −127
−11y − 80y − 400 = −127
−91y = 273
y = 3
Step 3: Find z
z = 4(3) + 20 = −8
Step 4: Find x
Substitute y = 3 and z = −8 into equation (1):
x + 2(3) + 3(−8) = 20
x + 6 − 24 = 20
x = 2
Final Solution
x = 2, y = 3, z = −8
Correct Answer
Option (C)
Conclusion
Using elimination and substitution methods, we solved the system of linear equations successfully.
Always verify the obtained values in all original equations to ensure accuracy.
