For what value of (k) the equation have a unique solution
2x+3y=1,kx+5y=7
Answers
Answer :
k ≠ 10/3
[ k € R - { 10/3 } ]
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 equations of straight lines are ;
2x + 3y = 1
kx + 5y = 7
The given equations can be rewritten as ;
2x + 3y - 1 = 0
kx + 5y - 7 = 0
Here ,
a = 2
a' = k
b = 3
b' = 5
c = -1
c' = -7
Now ,
For the given equations to have an unique solution a/a' ≠ b/b' .
Thus ,
=> a/a' ≠ b/b'
=> 2/k ≠ 3/5
=> 2 × (5/3) ≠ k
=> k ≠ 10/3
=> k € R - { 10/3 }
Hence ,
k can be any real number but not 10/3 .