I have two equations
First one is $\beta q_1 x_1 + q_2 x_2 =I-\gamma$
Second one is $q_1x_1+q_2 x_2=I$
where $\beta q_1\lt q_1$ and $I-\gamma\lt I$ and all are positive.
The slope of the first model is =-$q_2\over\beta q_1$
And the slope of the second model is =-$q_2\over q_1$
I think the first slope is greater than the second slope. Right? I am not sure. Please check the comparison again.
Then, how are their graphs? I posted below. First figure or second figure represents the equations ?
Which graph is true?
The second one seems more correct to me. Am I right?
The second one is correct!
Because $I - \gamma < I$, you get $\frac{I - \gamma}{q_2} < \frac{I}{q_2} $
That enables you to find the intersections with the $x_2$ axis like you have in the second drawing.
On the other hand, the slopes you found are wrong! Or at least the ones yoi wrote in your question. Please review the steps you did. Can you find the mistake?
As you ask, I will do the second equation:
$$q_1 x_1 + q_2 x_2 = I \iff q_2 x_2 = -q_1x_1 + I \iff x_2 = -\frac{q_1}{q_2}x_1 + \frac{I}{q_2}$$