If the system of equations
x - 2y + 3z = 9
2x + y + z = b
x - 7y + az = 24,
has infinitely many solutions, then a - b is equal to.........

Solution

Maths sir · 4 steps

Infinitely many solutions means the third equation must be a combination of the first two.

1. What infinite needs. For three equations to have infinitely many solutions, one of them must give no new information. That means the third equation is a combination of the first two, coefficients and right-hand side alike.
2. Match x and y. Try p(eq 1) + q(eq 2). Matching x gives p + 2q = 1. Matching y gives -2p + q = -7. Solving gives p = 3 and q = -1.

   3(x - 2y + 3z) - (2x + y + z) = x - 7y + 8z = 27 - b

3. Compare with eq 3. This must be the same equation as x - 7y + az = 24. So a = 8, and 27 - b = 24 gives b = 3. As a check, Δ = 5a - 40 = 0 at a = 8, and Δₓ = 15 - 5b = 0 at b = 3.
4. So. a - b = 8 - 3 = 5.

The idea to keep. For infinitely many solutions, write one equation as a combination of the others. Fix the multipliers using two variables, then match the rest. This is the same as Δ = Δₓ = Δ_y = Δ_z = 0.

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