Question

for what value of R,the pair of linear equations rx-y=2, 6x-2y=3 have on solution​

Answer 1

Answer :

r = 3

Note:

★ A linear equation in 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 pair of linear equations is ;

rx - y = 2 → rx - y - 2 = 0 ------(1)

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

Now ,

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

a = r

a' = 6

b = -1

b' = -2

c = -2

c' = -3

Now ,

a/a' = r/6

b/b' = -1/-2 = ½

c/c' = -2/-3 = ⅔

Now ,

The given pair of linear equations will have no solution if ,

=> a/a' = b/b' ≠ c/c'

=> r/6 = ½ ≠ ⅔

=> r/6 = ½

=> r = ½ × 6

=> r = 3

Hence , r = 3 .