If the system of equations
x+2 y-3 z=2 ; 2 x+λ y+5 z=5 ; 14 x+3 y+μ z=33
has infinitely many solutions, then λ+μ is equal to :

Solution

Maths sir · 4 steps

Eliminate x, and the two equations left must be the same equation, constant term included.

1. What infinite means. Three equations give infinitely many solutions here only when one of them adds nothing new. So after you eliminate x, the two remaining equations must be multiples of each other, constants and all.
2. Eliminate x. Second minus 2 times the first gives (λ-4)y + 11z = 1. Third minus 14 times the first gives -25y + (μ+42)z = 5.

   -25/(λ-4) = (μ+42)/11 = 5/1

3. Match the multiplier. The constants 1 and 5 fix the multiplier at 5. Then 5(λ-4) = -25 gives λ = -1, and μ + 42 = 55 gives μ = 13.
4. So. λ + μ = -1 + 13 = 12, which is option (c).

The idea to keep. For infinitely many solutions, eliminate one variable. The two equations left must be multiples of each other, and the constant terms tell you the multiplier.

Answer from the NTA final answer key. Solution by PrepWiser.