Question

find the value of k, such that the system of linear equation has a unique solution:

2x + 3y = 7 and kx + 9y = 15

Answer 1

2x+3y=7
kx+9y=15

cond for unique soln
a1/a2≠b1/b2
2/k≠3/9
2*9≠3*k
18≠3k
18/3≠k
6≠k

Answer 2

Concept

We will write the above system of linear equations in matrix form and will use row operations of matrix to obtaine the echelon form of the matrix. We know that if the system has a unique solution then the rank of the matrix is equal to the number of variables in the system of equations.

Given

The linear equations are given as

2x + 3y = 7

kx + 9y = 15

The number of variables are two.

Find

We have to calculate the value of k.

Solution

Writing the above equation in the matrix form, we have

$A = \left[\begin{array}{cc}2&3\\k&9\end{array}\right]$

$B = \left[\begin{array}{c}7\\15\end{array}\right]$

Therefore the augmented matrix will be,

$[A, B] = \left[\begin{array}{ccc}2&3&...7\\k&9&...15\end{array}\right]$

R2 → R2- k/2R1

$[A, B] = \left[\begin{array}{ccc}2&3&...7\\0&9-3k/2&15-7k/2\end{array}\right]$

Since rank of the matrix is 2 therefore,

9 - (3k/2) ≠ 0

k ≠ 6

and 15 - (7k/2) = 0

k = 30/7

k=  4.3

Hence the value of k is 4.3 or any other interger except 6.

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