# Algebraic Equations Questions and Answers – Solution of Linear Simultaneous Equation using Direct Methods

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This set of Numerical Methods Quiz focuses on “Solution of Linear Simultaneous Equation using Direct Methods”.

1. The problems which deal with the analysis of electronic circuits consisting of invariant elements depend on __________
a) The solution of simultaneous algebraic equations
b) Solution of transcendental equations
c) Interpolation problems
d) Finite difference method

Explanation: The problems which deal with the analysis of electronic circuits consisting of invariant elements depend upon linear equations with certain variables which can be solved by finding solution of simultaneous algebraic equations.

2. Which of the methods is direct method for solving simultaneous algebraic equations?
a) Jacobi’s method
b) Relaxation method
c) Cramer’s rule
d) Gauss seidel method

Explanation: Cramer’s rule is the direct method for solving simultaneous algebraic equations. In other methods, certain iterations are involved and that’s why the process becomes tedious and consequently indirect.

3. How many types of methods are there to solve simultaneous algebraic equations?
a) 2
b) 3
c) 4
d) 5

Explanation: There are two types of methods to solve simultaneous algebraic equations, namely, direct and iterative methods.

4. Direct methods are preferred over iterative methods as they provide solution faster.
a) True
b) False

Explanation: Iterative methods are preferred over direct methods as they provide solution faster. And, amount of accuracy depends upon accuracy desired which is achieved by iterative methods.

5. Which of the following method generally converges the solution?
a) Hit and trial method
b) Approximation method
c) Iterative method
d) Direct method

Explanation: Iterative methods generally converge the solution to get the results faster. With the increase in the number of iterations, the results tend to reach the correct solution.

6. What are the advantages of direct methods for solving the simultaneous algebraic equations?
a) Rounding of errors get propagated
b) Quite time consuming
c) Requires more recording of data
d) Yield a solution after a finite number of steps for any non-singular set of equations

Explanation: The only advantage of direct method is that we can yield a solution after a finite number of steps for any non-singular set of equations.

7. How many types of instabilities occur while solving a linear system of equations?
a) 2
b) 3
c) 4
d) 5

Explanation: There are two types of instabilities which occur while solving a linear system of equations, namely, induced and inherent stability.

8. What is induced instability?
a) Instability due to reduction in problem
b) Instability due to number of computations
c) Instability due to errors in programming
d) Instability due to incorrect choice of method for solving equations

Explanation: Induced instability is the Instability due to incorrect choice of method for solving equations. As different methods have different accuracy level.

9. What is inherent instability?
a) Instability due to convergence of solution
b) Instability due to ill conditioned set of equations
c) Instability due to propagation of rounding of errors
d) Instability due to incorrect choice of method of solution

Explanation: Inherent instability is the Instability due to ill conditioned set of equations. The linear system is called ill conditioned, if small changes in the coefficients of equations result in small changes in the values of the unknowns.

10. When the linear system is called ill conditioned?
a) If small changes in the coefficients of equations result in large changes in the values of the unknowns
b) If reduction is done in the problem
c) If the programming of linear system is expensive
d) If small changes in the coefficients of equations result in small changes in the values of the unknowns

Explanation: The linear system is called ill conditioned, if small changes in the coefficients of equations result in large changes in the values of the unknowns.

11. If approximate solution of the set of equations, 2x+2y-z = 6, x+y+2z = 8 and -x+3y+2z = 4, is given by x = 2.8 y = 1 and z = 1.8. Then, what is the exact solution?
a) x = 3 , y = 1 , z =2
b) x = 1 , y = 2 , z =2
c) x = 2 , y = 3 , z = 1
d) x = 1 , y = 3 , z = 2

Explanation: Substituting the approximate values x’ = 2.8, y’ = 1 z’ = 1.8 in the given equations, we get

2(2.8) + 2(1) – 1.8 = 5.8 …………..(i)
2.8 + 1 + 2(1.8) = 7.4 ……………..(ii)
-2.8 +3(1) + 2(1.8) = 3.8 …………..(iii)

Subtracting each equation (i), (ii), (iii) from the corresponding given equations we obtain

2xe + 2ye – ze = 0.2 …………….(iv)
Xe + ye +2ze = 0.6 ………………(v)
-xe + 3ye + 2ze = 0.2 ……………(vi)

Where xe = x – 2.8, ye = y – 1, ze = z = 1.8.
Solving the equations (iv), (v), (vi), we get xe = 0.2, ye = 0, ze = 0.2.
This gives the better solution x = 3, y = 1, z = 2, which incidentally is the exact solution.

12. 1.01x + 2y and x + 2y = 2 is well conditioned set of linear equations.
a) True
b) False

Explanation: Its solution is x = 1 and y = 0.5.
Now consider the system x + 2.01y = 2.04 and x + 2y = 2 which has the solution x = -6 and y = 4.
Hence the system is ill conditioned.

13. How can we avoid instability?
a) Reformulation of problem suitably
b) Making small changes in the coefficients of equations
c) Choosing the method involving higher computations
d) Rounding off the errors 