Question

find K if the pair of equations have no solutions 2x+ky=5 , and 6x+3y =9​

Answer 1

Answer :

k = 1

Note :

Note:

★ A linear equation is two variables represent a straight line .

★ The word consistent is used for the system of equations which consists any solution .

★ The word inconsistent is used for the system of equations which doesn't consists any solution .

★ Solution of a system of equations : It refers to the possibile values of the variable which satisfy all the equations in the given system .

★ A pair of linear equations are said to be consistent if their graph ( Straight line ) either intersect or coincide each other .

★ A pair of linear equations are said to be inconsistent if their graph ( Straight line ) are parallel .

★ If we consider equations of two straight line

ax + by + c = 0 and a'x + b'y + c' = 0 , then ;

• The lines are intersecting if a/a' ≠ b/b' .

→ In this case , unique solution is found .

• The lines are coincident if a/a' = b/b' = c/c' .

→ In this case , infinitely many solutions are found .

• The lines are parallel if a/a' = b/b' ≠ c/c' .

→ In this case , no solution is found .

Solution :

Here ,

The given linear equations are ;

2x + ky = 5 → 2x + ky - 5 = 0 ------(1)

6x + 3y = 9 → 6x + 3y - 9 = 0 ------(2)

Now ,

Comparing eq-(1) and eq-(2) with the general equations ax + by + c = 0 and a'x + b'y + c' = 0 respectively , we get ;

a = 2

a' = 6

b = k

b' = 3

c = -5

c' = -9

Thus ,

a/a' = 2/6 = 1/3

b/b' = k/3

c/c' = -5/-9 = 5/9

Also ,

It is given that , the pair of given equations has no solution , thus a/a' = b/b' ≠ c/c' .

Clearly , a/a' ≠ c/c' , thus sufficient condition for the given pair of equations to have no solution is a/a' = b/b' .

Thus ,

=> 1/3 = k/3

=> k = 1

Hence , k = 1 .