Why the indeterminate equation, which has coprime coefficients, gives the same solution when we set the value of one variable to equal the value of the other variable? I will clear it with an example: For instance, we have $$43x+47y=450$$ Now we put $x=y$.
Then $$43x+47x=450$$ Or, $$90x=450$$ Or $x=5$ Hence $x=y=5$ is a solution.