Consider the system of linear equations x + y + z = 2, 2x + 3y + 2z = 5, 2x + 3y + (a^{2 – 1})z = a + 1 then

a)
System has a unique solution for |a| = √3

b)
System is inconsistence for |a| = √3

c)
System is inconsistence for a = 4

d)
System is inconsistence for a = 3

Correct answer is option 'B'. Can you explain this answer?

Question

Consider the system of linear equations x + y + z = 2, 2x + 3y + 2z = 5, 2x + 3y + (a^{2 – 1})z = a + 1 then

a)
System has a unique solution for |a| = √3
b)
System is inconsistence for |a| = √3
c)
System is inconsistence for a = 4
d)
System is inconsistence for a = 3

Correct answer is option 'B'. Can you explain this answer?

Answer

We are given the system of linear equations:

x + y + z = 2 ...(1)
2x + 3y + 2z = 5 ...(2)
2x + 3y + (a^2) = b ...(3)

To find the value of a^2, we need to solve the system of equations.

First, let's subtract equation (1) from equation (2) to eliminate x:

(2x + 3y + 2z) - (x + y + z) = 5 - 2
x + 2y + z = 3 ...(4)

Now, let's subtract equation (4) from equation (3) to eliminate x:

(2x + 3y + (a^2)) - (x + 2y + z) = b - 3
x + y + (a^2) - 2y - z = b - 3
(a^2 - y - z) = b - 3 ...(5)

Now, we have two equations: (4) and (5). We can solve these equations to find the values of y and z.

From equation (4):
x + 2y + z = 3

From equation (5):
(a^2 - y - z) = b - 3

Rearranging equation (4), we get:
x = 3 - 2y - z ...(6)

Substituting equation (6) into equation (5), we get:
(a^2 - y - z) = b - 3

Substituting equation (6) into equation (5), we get:
(a^2 - y - z) = b - 3

Now, we have two equations in terms of y and z:

(a^2 - y - z) = b - 3 ...(7)
3 - 2y - z + 2y + z = 2 ...(8)

Simplifying equation (8), we get:
3 = 2

Since equation (8) is inconsistent, it means that the system of equations does not have a unique solution. Therefore, we cannot determine the value of a^2.

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