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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 - b)x + (a + b)y = 3a + b - 2
उत्तर
2x + 3y -7 = 0
(a - b)x + (a + b)y - 3a + b - 2 = 0
Here `a_1 = 2, b_1 = 3,c_1 = -7`
`a_2 = (a - b), b_2 = (a + b), c_2 = -(3a + b - 2)`
`a_1/a_2 = 2/(a - b), b_1/b_2 = 3/(a + b), c_1/c_2 = (-7)/(-(3a + b - 2)) = (-7)/(3a + b - 2)`
For the equation to have infinitely many solutions, we have:
`a_1/a_2 = b_1/b_2 = c_1/c_2`
`2/(a - b) = 7/(3a + b -2)`
6a + 2b - 4 = 7a - 7b
a- 9b = -4 ......(1)
`2/(a -b) = 3/(a + b)`
a - 5b = 0 .....(2)
Subtracting (1) from (2), we obtain
4b = 4
b = 1
Substituting the value of b in equation (2), we obtain
a - 5 x 1 = 0
a = 5
Thus, the values of a and b are 5 and 1 respectively.
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