Find all values of h and k such that the system has

Question

no solutions, unique solutions and infinite solutions?

hx +6y = 2 
x +(h+1)y =2k 

Answer 1

The equations are hx + 6y = 2 ---> (1) and x + y(h + 1) = 2k ---> (2)

Write the equations into matrices form AX = B.

A is coefficient matrix,X is variable matrix and B is constant matrix.

DetA = h(h + 1) - 6(1) = h² + h - 6

Case 1: If the determinant is non zero, then the slopes must be different and the lines must be intersect in exactly one point.

det A ≠ 0 then the system has unique solution.

h² + h - 6 ≠ 0

h² + 3h - 2h - 6 ≠ 0

h(h + 3) - 2(h + 3) ≠ 0

h + 3 ≠ 0 and h - 2 ≠ 0

h ≠ - 3 and h ≠  2

If h ≠ -3 and h ≠ 2 then the system has unique solution.

The augmented matrix is

Apply the row elementary operations to put the matrix into reduced row-echelon form.

Case 2 :  If a row reduced matrix has row of zeros on the left side, but right hand side isn't not zero.There is no solution.

Comments

Contd...

Case 3 :  If a row reduced matrix has row of zeros on the left side and right hand side then there is infinite solutions.

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