for what value of a, the pair if linear equation. ax+3y=a-3,12x+ay=6 to represent coincident lines
Answers
Answer :
a = 6
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 ;
ax + 3y = a - 3 => ax + 3y + 3 - a --------(1)
12x + ay = 6 => 12x + ay - 6 = 0 --------(2)
Clearly , we have ;
a = a
a' = 12
b = 3
b' = a
c = 3 - a
c' = -6
Now ,
For the given lines to be coincident ,
a/a' = b/b' = c/c'
Thus ,
=> a/12 = 3/a = (3 - a)/-6
=> a/12 = 3/a = (a - 3)/6
Case1
• Considering a/12 = 3/a
=> a/12 = 3/a
=> a×a = 3×12
=> a² = 36
=> a = √36
=> a = ±6
=> a = 6 , -6
Case2
• Considering 3/a = (a - 3)/6
=> 3/a = (a - 3)/6
=> 3×6 = a(a - 3)
=> 18 = a² - 3a
=> a² - 3a - 18 = 0
=> a² - 6a + 3a - 18 = 0
=> a(a - 6) + 3(a - 6) = 0
=> (a - 6)(a + 3) = 0
=> a = 6 , -3
°•° The common value of a in both the cases is 6 .
•°• a = 6 is the appropriate value .