Question

If the system of equations \[ x-3y+5z=3 \] \[ x-2y+4z=4 \] \[ 2x-7y+\lambda z=5 \] has infinite number of solutions, then the value of \(\lambda\) is:

• A. \(2\)
• B. \(4\)
• C. \(5\)
• D. \(11\)

Hint:

A system has infinitely many solutions when one equation is dependent on the others.

Answer 1

The Correct Option is D

Step 1: Write coefficients in ratio form For infinite number of solutions: \[ \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=\frac{d_1}{d_2} \] after reducing dependent equations. Observe: \[ 2x-7y+\lambda z=5 \] can be obtained by combining first two equations.
Step 2: Subtract second equation from first equation \[ (x-3y+5z)-(x-2y+4z)=3-4 \] \[ -y+z=-1 \] \[ y-z=1 \]
Step 3: Use first equation From: \[ y-z=1 \] \[ y=z+1 \] Substitute into: \[ x-2y+4z=4 \] \[ x-2(z+1)+4z=4 \] \[ x+2z=6 \] \[ x=6-2z \] Thus infinitely many solutions are possible if third equation is satisfied automatically.
Step 4: Substitute into third equation \[ 2x-7y+\lambda z=5 \] Substitute: \[ x=6-2z,\qquad y=z+1 \] \[ 2(6-2z)-7(z+1)+\lambda z=5 \] \[ 12-4z-7z-7+\lambda z=5 \] \[ 5+(\lambda-11)z=5 \] For infinitely many solutions: \[ (\lambda-11)z=0 \] for all \(z\). Hence: \[ \lambda-11=0 \] \[ \lambda=11 \] Option analysis:
• Option (A): Incorrect
• Option (B): Incorrect
• Option (C): Incorrect
• Option (D): Correct Therefore: \[ \boxed{\text{(D)}} \]