if piar a linear equation x+my+4=0 and 2x+5y-3=0are consistent then. find the value of m. please
Answers
Answer:
m ≠ 5/2
ie ; m can be any real number other than 5/2
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 system of linear equations are ;
x + my + 4 = 0
2x + 5y - 3 = 0
Clearly,
a = 1 , b = m , c = 4
a' = 2 , b' = 5 , c = -3
Thus,
a/a' = 1/2
b/b' = m/5
c/c' = 4/-3 = -4/3
Also,
According to the question , the given system of linear equations is consistent .
Thus,
The lines may be either coincident or intersecting .
★ Suppose the lines are coincident :
Then ,
a/a' , b/b' and c/c' must be equal .
ie; a/a' = b/b' = c/c'
But here , a/a' ≠ c/c'
Thus,
The lines can't be coincident .
★ Suppose the lines are intersecting :
Then ,
a/a' and b/b' must not be equal .
ie ; a/a' ≠ b/b'
=> 1/2 ≠ m/5
=> m/5 ≠ 1/2
=> m ≠ 5/2
Thus,
The lines are intersecting if m ≠ 5/2