3.3: Solution of the Homogeneous equation ax+by = 0

Proposition 3.5

The general solution of the homogeneous equation \(r_{1}x+r_{2}y = 0\) is given by

\[\begin{array} {ccc} {x = k \frac{r_{2}}{\gcd(r_{1}, r_{2})}}&{and}&{y = -k \frac{r_{2}}{\gcd(r_{1}, r_{2})}} \nonumber \end{array}\]

where \(k \in \mathbb{Z}\).

Proof

On the one hand, by substitution the expressions for \(x\) and \(y\) into the homogeneous equation, one checks they are indeed solutions. On the other hand, \(x\) and \(y\) must satisfy

\[\frac{r_{1}}{\gcd(r_{1}, r_{2})} x = - \frac{r_{2}}{\gcd(r_{1}, r_{2})} y \nonumber\]

The integers \(r_{i}\) (for \(i\) in \(\{1,2\}\)) have greatest common divisor equal \(\gcd(r_{1},r_{2})\) to \(1\). Thus Euclid’s lemma applies and therefore \(\frac{r_{1}}{\gcd(r_{1}, r_{2})}\) is a divisor of \(y\), while \(\frac{r_{2}}{\gcd(r_{1}, r_{2})}\) is a divisor of \(x\).

Corollary 3.7

Let \(\{b_{i}\}^{n}_{i=1}\) be non-zero integers and denote \(B = \mbox{lcm} (b_{1}, \cdots b_{n})\). The general solution of the homogeneous system of equations \(x = _{b_{i}} 0\) is given by

\[x =_{B} 0 \nonumber\]

Proof

From the definition of \(\mbox{lcm} (b_{1}, \cdots, b_{n})\), every such \(x\) is a solution. On the other hand, if \(x \ne _{B} 0\), then there is an \(i\) such that \(x\) is not not a multiple of \(b_{i}\), and therefore such an \(x\) is not a solution.