If the system of linear equations
2x - 3y = γ + 5,
α x + 5y = β + 1, where α, β, γ ∈ R has infinitely many solutions then the value
of | 9α + 3β + 5γ | is equal to ____________.

Solution

Maths sir · 4 steps

Infinitely many solutions means the two lines coincide, so every coefficient must be in the same ratio.

1. Coincident lines. Two equations in two unknowns have infinitely many solutions only when they describe the same line. Then the x-coefficients, the y-coefficients and the constants are all in one common ratio.

   2/α = -3/5 = (γ + 5)/(β + 1)

2. Find alpha. From 2/α = -3/5 you get α = -10/3, so 9α = -30.
3. Link beta and gamma. From (γ + 5)/(β + 1) = -3/5, cross-multiply: 5γ + 25 = -3β - 3. This gives 3β + 5γ = -28, so you never need β and γ separately.
4. So. 9α + 3β + 5γ = -30 - 28 = -58, and its absolute value is 58.

The common slip: -58. Forgetting the modulus gives -58. The question asks for |9α + 3β + 5γ|, which is 58.

The idea to keep. For two lines, infinitely many solutions means all three ratios are equal. If the question asks for a combination of unknowns, look for it directly in the ratio equations instead of solving for each one.

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