For the system of equations ax + by = 2 and 3x - (5 - 2a)y = 6, find the values of a and b such that the system has infinitely many solutions.

Question

Accepted Answer

For two linear equations to have infinitely many solutions, their coefficients must be proportional, and the constants must also be proportional.

Given equations:
1) ax + by = 2
2) 3x - (5 - 2a)y = 6

Rewrite second equation:
3x - (5 - 2a)y = 6
=> 3x + [- (5 - 2a)]y = 6
=> 3x + (2a - 5)y = 6

For infinite solutions:
(a / 3) = (b / (2a - 5)) = (2 / 6)

Since 2/6 = 1/3, we have:
(a / 3) = (b / (2a - 5)) = 1/3

From (a / 3) = 1/3 => a = 1

From (b / (2a - 5)) = 1/3
Substitute a = 1:
(b / (2*1 - 5)) = 1/3
(b / (2 - 5)) = 1/3
(b / -3) = 1/3
b = -3 * (1/3) = -1

Therefore, a = 1 and b = -1.

Hence, option 1 is correct.

Answer

Answer

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