Question

14. Find the values of a and b for which the following system of equations has infinite number of solutions:

5x + 3y = 15 ;

( a + b )x + ( 4a + b )y = 1

Standard:- 10

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Answer 1

Given Equation is 5x + 3y = 15 

= > 5x + 3y - 15 = 0 ------ (1)

Given Equation is (a + b)x + (4a + b)y = 1  ------ (2)

a1 = 5, b1 = 3, c1 = -15

a2 = (a + b), b2 = (4a + b), c2 = -1


Now,

Given that the Equation has the infinite number of solutions.

= > (a1/a2) = (b1/b2) = (c1/c2)

= > (5/a + b) = (3/(4a + b)) = (-15/-1)

(1)


= > (5/a + b) = 15

= > 5 = 15(a + b)

= > a + b = (1/3)    --------------- (3) 




(2) 

= > (3/4a + b) = 15

= > 3 = 15(4a + b)

= >  4a + b = (1/5)   - ---------- (4) 


On solving (3) * 4 & (4), we get

= > 4a + 4b = (4/3)

= > 4a + b = (1/5)

      ----------------------

              3b = 17/15

                 b = 17/45.



Substitute b = (17/45) in (3), we get

= > a + b = (1/3)

= > a + (17/45) = (1/3)

= > a = (1/3) - (17/45)

= > a = (-2/45).





Therefore the value of a = -2/45 and b = 17/45.



Hope this helps!