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प्रश्न
Find the values of a and b for which the following system of equations has infinitely many solutions:
2x + 3y = 7
(a - 1)x + (a + 1)y = (3a - 1)
उत्तर
The given system of equations is
2x + 3y - 7 = 0
(a - 1)x + (a + 1)y - (3a - 1) = 0
It is of the form
`a_1x + b_1y + c_1 = 0` `
a_2x + b_2y + c_2 = 0`
Where `a_1 = 2, b_1 = 3, c_1 = -7`
And `a_2 = a - 1, b_2 = a + 1, c_2 = -(3a - 1)`
The given system of equations will be have infinite number of solutions, if
`a_1/a_2 = b_1/b_2 = c_1/c_2`
`=> 2/(a - b) = 3/(a + 1) = (-7)/(-(3a - 1))`
`=> 2/(a - 1) = 3/(a + 1) = (-7)/(-(3a - 1))`
`=> 2/(a - 1) = 3/(a + 1) = (-7)/(3a -1)`
`=> 3/(a - 1) = 3/(a + 1) and 3/(a + 1) = 7/(3a - 1)`
=> 2(a + 1) = 3(a - 1) and 3(3a - 1) = 7(a + 1)
=>2a + 2 = 3a - 3 and 9a - 3 = 7a + 7
=> 2a - 3a = -3 and 9a - 3 = 7a + 7
=> -a = -5 and 2a = 10
=> a = 5 and a = 10/2 = 5
=> a =5
Hence, the given system of equations will have infinitely many solutions
if a = 5
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