If $m= a_1x+b_1y$, $n= a_2x + b_2y$ and $a_1b_2-a_2b_1=1$ then prove that $(m,n)=(x,y)$

Question

$$ \begin{cases} m=&a_1x+ b_1y \\ n =& a_2x + b_2y \end{cases} \qquad\text{and}\qquad a_1 b_2 - a_2 b_1 = 1 $$

I tried to substitute the value of $x$ in eq-1 to eq-2 and got $na_1+ma_2=y$

source: Challenges and Thrills Of Pre-College Mathematics (Exercise 2.2 Q-20)

You made a mistake, that's not the value $y$. – jjagmath Apr 07 '21 at 12:45 — jjagmath, Apr 07 '21 at 12:45

score 0 · Accepted Answer · answered Apr 07 '21 at 12:40

0

We have $(x,y)$ divides every linear combination of $x$ and $y$, so clearly $(x,y)$ divides $m$ and $n$. Then it divides $(m,n)$.

Now solve for $x$ and $y$ in terms of $n$ and $m$ and repeat the argument.

answered Apr 07 '21 at 12:40

jjagmath

i did not usnderstand your last statement did u want to say that i solve for x from eq1 and eq 2? – Chaitanya Bhoite Apr 07 '21 at 14:30
Yes, solve the system of equations to get $x$ and $y$ as linear combinations of $m$ and $n$. – jjagmath Apr 07 '21 at 16:16

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