For the following equations, what are the values of a and b to have infinitely many solutions? ax + by = 2 3x - (5 - 2ay) = 6
a = 1, b = -1
To have infinitely many solutions, the two equations must be identical (i.e., one equation should be a multiple of the other). Let's work with the equations:
First equation: ax + by = 2.
Second equation: 3x - (5 - 2ay) = 6, which simplifies to 3x + 2ay - 5 = 6, or 3x + 2ay = 11.
For the two equations to be identical, the ratios of the coefficients of x, y, and the constant terms should be equal. That is, the ratio of coefficients of x (a/3), the ratio of coefficients of y (b/2a), and the ratio of constants (2/11) must be the same.
Solving these equations gives a = 1 and b = -1.
Thus, the correct answer is option 1: a = 1, b = -1.
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