If $Ax+By=c$ represents a linear quation. What is $c$ in this equation and what does it represent intuitively?

Question

When we write down the simultaneous equations in the form of $Ax=b$ what does the elements in vector $b$ tell us intuitively?

Answer 1

For the question in the title, we have $$Ax+By=C\implies y=\frac CB-\frac AB x$$ so $C/B$ is the intercept. The point $(0, C/B)$ is precisely where the line crosses the $y$-axis.

Say we have the system of equations $$\alpha x+\beta y=\gamma\\\delta x+\epsilon y=\theta$$ yielding the form $$\begin{bmatrix}\alpha&\beta\\\delta&\epsilon\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}\gamma\\\theta\end{bmatrix}.$$ Note that the vector $b=[\gamma\quad\theta]^T$ does not have much meaning in itself.

Answer 2

In the equation of a line in standard form. $a x + by = c.$ $c$ is a scaled distance from the origin. $\frac {|c|}{\sqrt {a^2+b^2}}$ is the actual distance from the origin.

As far as what $b$ means intuitively in a system of equations $Ax = b,$ where $x$ and $b$ are vectors. I don't have much for you.

To some degree $\|b\|$ has some relation to $\|x\|$ but it depends on the magnitude of the eigenvalues of $A.$ So, that is not much to go on.