Consider the following system 3x1 5x2 h 6x1 kx2 8 Determ

Consider the following system:

3x₁ + 5x₂ = h

-6x₁ + kx₂ = 8

Determine all values of h and k for which system has:

a) a unique solution

b) no solutions

c) infinitely many solutions

Solution

We have been given the following equations:

3x₁ + 5x₂ = h ......(1) and

-6x₁ + kx₂ = 8.... (2)

On multiplying the 1st equation by 2 and adding the result to the 2nd equation, we get, (10 + k) x₂ = 2h + 8 so that

x₂ = (2h + 8)/ (10 + k)

Also on multiplying the 1st equation by k and the 2nd equation by 5 and then subtracting the 2nd resulting expression from the 1st, we get (3k + 30) x₁ = hk-40 or, x₁ = (hk-40)/ 3(k+10)

We have thus determined x₁ and x₂ in terms of h ank. For each value of h and k ( when k is not equal to -10), x₁ and x₂ will have unique solutions.

We know that division by 0 gives an indeterminate result. Thus when k = -10, 10 + k = 0 and then the given system of linear equations has no solution.

We also know that only one equation in 2 variables gives infinitely many solutions. Thus, when h = -4 and k = -10, both the equations become identical leading to only one equation in 2 variables which gives infinitely many solutions.