Q.1 Consider the system of linear equations
x+2y+z = 3
2x+7y+az = b
ay+5z = 10
Find those pair of values of (a,b) for which the system has infinitely many solutions.
Answers
Given : x+2y+z = 3 , 2x+7y+az = b , ay+5z = 10
To find : pair of values of (a,b) for which the system has infinitely many solutions.
Solution:
x+2y+z = 3 Eq1
2x+7y+az = b Eq2
ay+5z = 10 Eq3
Eq2 - 2 * Eq1
=> 3y + (a - 2)z = b - 6
ay + 5z = 10
These will have infinite Solutions
is 3/a = (a - 2)/5 = (b - 6)/10
=> 15 = a² - 2a
=> a² - 2a - 15 = 0
=> a² - 5a + 3a - 15 = 0
=> a(a - 5) + 3(a - 5) = 0
=> (a + 3)(a - 5) = 0
=> a = - 3 , a = 5
Case 1 , a = - 3
=> 3/a = (a-2)/5 = - 1
=> - 1 = (b - 6)/10
=> -10 = b - 6
=> b = - 4
(-3 , - 4)
Case 2 , a = 5
=> 3/a = (a-2)/5 = 3/5
=> 3/5 = (b - 6)/10
=> 6 = b - 6
=> b = 12
( 5 , 12)
(-3 , - 4) & ( 5 , 12) for which the system has infinitely many solutions.
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