For what value of k, the following system of equations kx+2y=3, 3x+6y=10 a unique solution.
Answers
Answer :
k ≠ 1
( k can be any real number other than 1 )
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 ;
• kx + 2y = 3 → kx + 2y - 3 = 0
• 3x + 6y = 10 → 3x + 6y - 10 = 0
Clearly , we have ;
a = k
a' = 3
b = 2
b' = 6
c = -3
c' = -10
Now ,
For the given linear equations to have an a unique solution , we have ;
=> a/a' ≠ b/b'
=> k/3 ≠ 2/6
=> k/3 ≠ 1/3
=> k ≠ 3×⅓
=> k ≠ 3/3
=> k ≠ 1