Recently, one of my friends have told me that the following two equations are equivalent on the basis of the number of solutions.
I checked the number of solutions to the two equations and found that his comment is right i.e the number of solutions to the two equations is equal. My question is that why the number of solutions to the two equations is equal? Can anyone explain me the reasoning behind the occurrence of such thing?
On
Define $y_1=x_1-2$, $y_2=x_2+2$, $y_3=x_3$, $y_4=x_4-3$. Now the first inequality for the $x's$ gives $$2\leq x_1\leq 6\Leftrightarrow 2-2\leq x_1\leq 6-2\Leftrightarrow 2\leq y_1\leq 4$$ So with this you get the first inequality needed. The same idea works with the others. Now if you sum the $y's$, you get: $$y_1+y_2+y_3+y_4=(x_1-2)+(x_2+2)+(x_3)+(x_4-3)=x_1+x_2+x_3+x_4-3=15-3=12$$ This shows that the first system implies the second, reciprocally you can show that the second implies the first, so they are equivalent.
Hint
$$x_1+x_2+x_3+x_4=(x_1-2)+2+(x_2+2)-2+(x_3-0)+0+(x_4-3)+3$$ Now, change variables and bounds