Question

If the system of linear equations

Hint:

For infinitely many solutions, the coefficient matrix must be singular and the augmented system must be consistent.

Answer 1

Correct Answer: 3

Step 1: Write the coefficient matrix.
The coefficient matrix is
\[ A= \begin{bmatrix} 1 & 1 & 1\\ 1 & 2 & -1\\ a & 7 & 1 \end{bmatrix} \] For infinitely many solutions, the coefficient matrix must be singular.
So,
\[ \det(A)=0 \]

Step 2: Compute the determinant.
\[ \det(A)= \begin{vmatrix} 1 & 1 & 1\\ 1 & 2 & -1\\ a & 7 & 1 \end{vmatrix} \] \[ =1(2\cdot 1-(-1)\cdot 7)-1(1\cdot 1-(-1)\cdot a)+1(1\cdot 7-2a) \] \[ =9-(1+a)+(7-2a) \] \[ =15-3a \] For singularity,
\[ 15-3a=0 \] \[ a=5 \]

Step 3: Use consistency condition.
When \(a=5\), the third equation has coefficients
\[ (5,7,1) \] Now observe that
\[ (5,7,1)=3(1,1,1)+2(1,2,-1) \] Therefore, for consistency, the right hand side must also satisfy the same relation.
So,
\[ b+1=3(1)+2(b) \] \[ b+1=3+2b \] \[ b=-2 \]

Step 4: Find \(a+b\).
\[ a+b=5+(-2) \] \[ a+b=3 \]

Step 5: Final conclusion.
Hence,
\[ \boxed{3} \]