We know that points $(x_i,y_i)$ which satisfy the equation $ax_i+by_i=c$ lie on the same straight line. I understand that all points on this line satisfy the equation, but how do we ensure that all points satisfying the equation stay on the same line?
As @saulspatz says,
How do we know that the lines defined by analytic geometry satisfy the Euclidean axioms?
$$ax+by=c$$
If $a \ne 0$, then we have $x =\frac{c-by}a$. Knowing $y$ completely determines $x$. Let $y$ takes value $t$
$$\begin{bmatrix} x\\ y \end{bmatrix}=\begin{bmatrix} \frac{c}{a}\\0\end{bmatrix} + t\begin{bmatrix} -\frac{b}{a}\\ 1\end{bmatrix}$$
The locus is parallel to the direction of $\begin{bmatrix} -\frac{b}{a} \\ 1\end{bmatrix}$ and passes through the point $\begin{bmatrix} \frac{c}{a}\\0\end{bmatrix} $. Since it moves along a direction, it is a line.
If $a=0$ and $b\ne 0$, then $y=\frac{c}{b}$ and $x$ can takes any values, this is clearly a horizontal line.