For what values of p the system of equations 4x+py+5=0 and 2x+3y+7=0 has exactly one solution? pls return mai plzz
Answers
Answer :
p ≠ 6 ie. p can be any real number other than 6 .
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 system of linear equations is ;
4x + py + 5 = 0 ------(1)
2x + 3y + 7 = 0 ------(2)
Now ,
Comparing the given equations (1) and (2) with the general linear equations ax + by + c = 0 and a'x + b'y + c' = 0 , we have ;
a = 4
a' = 2
b = p
b' = 3
c = 5
c' = 7
Also ,
We know that , for exactly one solution (ie. unique solution) a/a' ≠ b/b' .
=> 4/2 ≠ p/3
=> 2 ≠ p/3
=> 2•3 ≠ p
=> p ≠ 6