Find the value of a and B for which the following pair of linear equations has infinite number of solutions: 2x + 3y = 7; 8x + (a + B)y = 28
Answers
Step-by-step explanation:
Given :-
The pair of linear equations 2x + 3y = 7;
8x + (a + B)y = 28 has infinite number of solutions.
Correction :-
8ax + (a + B)y = 28
To find :-
Find the values of a and B ?
Solution :-
Given pair of linear equations in two variables are
2x + 3y = 7
=> 2x+3y -7 = 0
On comparing with a1x+b1y+c1 = 0 then
a1 = 2
b1 = 3
c1 = -7
and
8ax + (a + B)y = 28
=> 8ax +(a+B)y -28 = 0
On comparing with a2x+b2y+c2 = 0 then
a2 = 8a
b2 = (a+B)
c2 = -28
Now,
a1/a2 = 2/(8a) = 1/(4a)
b1/b2 = 3/(a+B)
c1/c2 = -7/-28 = 1/4
Given that
The given pair of linear equations in two variables have infinite number of solutions.
We know that
a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 have infinitely number of solutions, if a1/a2 = b1/b2 = c1/c2
=> 1/4a= 3/(a+B) = 1/4
On taking 1/(4a) = 1/4
=> 4a = 4
=> a = 4/4
=> a = 1
On taking 3/(a+B) = 1/4
On applying cross multiplication then
=> (a+B) = 3×4
=> a+B = 12
=> 1+B = 12
=> B = 12-1
=> B = 11
therefore, a = 1 and B = 11
Answer:-
The values of a and B are 1 and 11 respectively.
Used formulae:-
→
a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 have infinitely number of solutions, if a1/a2 = b1/b2 = c1/c2
Note :-
If the given equations are 2x + 3y = 7;
8x + (a + B)y = 28 has infinite number of solutions then we get only a+B = 12 .
We can't get the values of a and B individually.