Question

If the system of equations  
x + y + az = b  
2x + 5y + 2z = 6  
x + 2y + 3z = 3  
Has infinitely many solutions, then 2a + 3b is equal to

• A. 20
• B. 23
• C. 25
• D. 28

Answer 1

The Correct Option is B

To determine the value of \(2a + 3b\) for the given system of equations to have infinitely many solutions, we analyze the system:

• Equation 1: \(x + y + az = b\)
• Equation 2: \(2x + 5y + 2z = 6\) 
• Equation 3: \(x + 2y + 3z = 3\)

For a system of equations to have infinitely many solutions, the equations must be consistent and dependent. The rank of the augmented matrix should be equal to the rank of the coefficient matrix and less than the number of variables involved.

The coefficient matrix for this system is:

1	1	a
2	5	2
1	2	3

Performing row operations to achieve a row echelon form:

• Subtract row 1 from row 3:
  \( [0, 1, 3-a] \)
• Subtract twice row 1 from row 2:
  \( [0, 3, 2-2a] \)
• The reduced coefficient matrix is:

1	1	a
0	3	2-2a
0	1	3-a

For consistency and dependency, the last two equations must be proportional:

\( k(0, 1, 3-a) = (0, 3, 2-2a) \)

Equating corresponding terms gives:

• From second terms: \( k \times 1 = 3 \) → \( k = 3 \)
• From third terms: \( k(3-a) = 2-2a \)
• Substituting \( k = 3 \): \( 3(3-a) = 2-2a \)

Simplifying gives: \(9 - 3a = 2 - 2a\)

Rearranging terms, \(9 - 2 = 3a - 2a \) gives \( 7 = a \).

Substituting \( a = 7 \) into equation 1:

\(x + y + 7z = b\)

For infinite solutions, the equations are consistent:

• Substituting in equation 3, \( x + 2y + 3z = 3 \)

Substituting \( a = 7 \) is consistent with the determinant zero condition. Hence:

• The solution is given by: \( 2a + 3b = 2(7) + 3(3) = 14 + 9 = 23 \).

Therefore, the value of \( 2a + 3b \) is 23.