If 'a' and 'b' are real numbers, for what values does the equation 3x - 5 + a = bx + 1 have a unique solution 'x'?

For all 'a' and 'b'.

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For no 'a' and 'b'.

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For a

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For b

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Solution

The correct option is D For b 3.
Let two linear equations in single variable be $a_{1} x + b_{1} a n d a_{2} x + b_{2}$
So, for obtaining a unique solution:
$a_{1} \neq a_{2}$
This can be obtained as:
$a_{1} x + b_{1} = a_{2} x + b_{2} \Rightarrow (a_{1} - a_{2}) x = (b_{2} - b_{1}) \Rightarrow x = \frac{(b_{2} - b_{1})}{(a_{1} - a_{2})}$
So for x to be a proper value, $(a_{1} - a_{2})$ should not be zero.
So $a_{1} \neq a_{2}$
Applying the criteria for a unique solution:
$E q n .1 \Rightarrow 3 x - 5 + a \Rightarrow a_{1} = 3 E q n .2 \Rightarrow b x + 1 \Rightarrow a_{2} = b A s, a_{1} \neq a_{2} S o, b \neq 3.$

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