Is x Necessarily Rational If It Satisfies (ax+b)/(cx+d)=1?

[DarkGuju]

Suppose a,b,c,d are integers and a DOES NOT equal c. Suppose that x is a real number that satisfies the equation:

(ax+b)/(cx+d)=1

Must x be rational? If so, express x as a ratio of two integers.

I have no idea how to begin this problem.

 

[epenguin]

1. express this as x = (work it out)

2. See if you can apply the definition of 'rational' and get a conclusion

3. Something else - I'll tell you if and when you come back.

 

[DarkGuju]

(ax+b)=(cx+d)

ax-cx=d-b

x(a-c)=d-b

x=(d-b)/(a-c)

So if a,b,c,d are integers then the subtraction and division of integers must also be rational. Is that the answer?

 

[yyat]

(ax+b)=(cx+d)

ax-cx=d-b

x(a-c)=d-b

x=(d-b)/(a-c)

So if a,b,c,d are integers then the subtraction and division of integers must also be rational. Is that the answer?

Indeed it is!

 

[epenguin]

So now I will say point 3.

You said you had 'no idea how to begin'. So now you have done it, think about why you had no idea and how you did solve it. It was a question about x. So maybe it was a good idea to know what x was. Also often enough there are very few things you can do, so might as well try those. Also when asked whether something is rational, real, integral, prime, or other kind of number or thing, it can sound very abstruse till you ask yourself what that means, what is? a rational number for instance. The definition gave you the answer fairly directly. So now, having given yourself permission so to speak to do this actively instead of saying I have no idea and a blank page you have got yourself started and I'm sure will be able to handle the next problem actively and hopefully many others.