If the system of equations
2 x+7 y+λ z=3 ; 3 x+2 y+5 z=4 ; x+μ y+32 z=-1
has infinitely many solutions, then (λ-μ) is equal to ______ :

Solution

Maths sir · 4 steps

Start with D_z: it has no λ in it, so it gives μ straight away.

1. Condition to use. For infinitely many solutions, D = 0 and also Dₓ = D_y = D_z = 0. For D_z, replace the z-column with the constants 3, 4, -1. That removes λ, leaving only μ.

   D_z = |2, 7, 3; 3, 2, 4; 1, μ, -1| = 2(-2-4μ) - 7(-3-4) + 3(3μ-2) = μ + 39

2. Find mu. D_z = 0 gives μ = -39.
3. Find lambda. Put μ = -39 into D = |2, 7, λ; 3, 2, 5; 1, -39, 32|. Expanding along the first row gives 2(64+195) - 7(96-5) + λ(-117-2). That is 518 - 637 - 119λ = -119(1+λ). Setting D = 0 gives λ = -1, and Dₓ = 0 also checks out with these values.
4. So. λ - μ = -1 - (-39) = 38.

The common slip: -38. Working out μ - λ instead of λ - μ gives -38. Check the order the question asks for.

The idea to keep. With two unknown parameters, look for the determinant that drops one of them. Replacing that parameter's column with the constants gives the other one in a single step.

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