If $ax + by = a(b-1) + b(-1)$, then does $x = b-1$ and $y = -1$

Question

In this case, $x$ and $y$ are variables and $a$ and $b$ are arbitrary constants. It seems like just looking at the equation that this would be true, but is there a case when it does not work? If I try solving specifically for $x$ and $y$, I don't get that $x = b - 1$ and $y = -1$, but I probably need a second equation to be able to solve for the variables.

Answer 1

Its an equation with variable(s), you find the case(s) were it works. For an equation of 0 variables it's either is true or isn't. For an equation of 1 variable it works for one case, or for multiple cases. For an equation of more than 2 variables there will generally be multiple cases. You can either add another equation to create a unique solution or solve for x or y in terms of y or x.

Answer 2

This equation has infinitely many solutions of which one is the one that you mentioned. Unless further constraints are put on $x$ and $y$ we may as well say that $y=(a-1)$ and $x=-1$ is a solution.

Answer 3

You can rewrite (assuming $b\ne0$)

$$y= \frac{a(b-1-x) + b(-1)}b$$

and get an infinity of $(x,y)$ pairs, among which $(b-1,-1)$.

Answer 4

You can analyze the equation \begin{equation}\label{eqn:1} \tag{1}ax+by=a(b-1)+b(-1) \end{equation} by cases depending on the values of the constants $a$ and $b$.

First case: $a=0$ and $b=0$. Then the solutions for the equation \eqref{eqn:1} are $x$ any real number and $y$ any real number.

Second case: $a\neq0$ and $b=0$. Then the solutions for the equation \eqref{eqn:1} are $x=-1$ and $y$ any real number.

Third case: $a$ any number and $b\neq0$. Then the solutions for the equation \eqref{eqn:1} are $$y=\frac{ab-a-b}{b}-\frac{a}{b}x,$$ with $x$ any real number.

As you see, equation \eqref{eqn:1} has infinitely solutions. Then, the solution that you propose ($x=b-1$ and $y=-1$) is just one of them, and this solution will be obtained from the third case.

Finally, if you want to obtain a unique solution, effectively, you need a second equation which represents a line not parallel to the line given by equation \eqref{eqn:1}.