I'm working through Jim Hefferon's Linear Algebra. I was working on question 1.30 which reads as follows:
Prove that, where $a, b, c, d, e$ are real numbers with $a \ne 0$, if the linear equation
$ax + by = c$ has the same solution set as $ax + dy = e$ then they are the same equation. What if $a = 0$?
I attempted this problem for hours, but I was getting nowhere and ended up looking at the solution. The solution involves solving for $x$ in terms of $y$, which I understand is only possible because $a \ne 0$, and substituting a value of $y=0$. Why are we allowed to substitute a value of $y=0$ in this system of equations? How do we know that $y=0$ is part of the solution set of the system of equations? Is it the case that we're allowed to substitute any value we want in a system of equations so long as it doesn't cause a contradiction or division by $0$?
Best Answer
In this case, both equations provide a single solution $(x,y)$ for every value of $y$, because the steps that lead to $$x=\frac{c-by}a$$ are reversible.
So, both equations will have a unique solution of the form $(x_0,0)$ (i.e. when $y=0$), and thus the two values for $x_0$ must coincide by the hypothesis.