Difference between -x+y=3 and -4x+4y=12 lines equations

Question

I would like to know whats the difference between these two lines. My question is what it means to multiplicate the line equation by 4, what does it change? My professor said it has something to do with the ´line speed´

Answer 1

Let \begin{align} &l_1 \ldots\ -x+y = 3,\tag{1}\\ &l_2\ldots\ -4x+4y = 12\tag{2}. \end{align} Now, let point $(x_0,y_0)$ lie on the line $l_1$. Then, it satisfies the equation $(1)$, which means that $-x_0+y_0 = 3$. Multiplying by $4$ we get $-4x_0 + 4y_0 = 12,$ so the point $(x_0,y_0)$ satisfies the equation $(2)$ and therefore it lies on the line $l_2$. So, every point on $l_1$ also lies on $l_2$.

Conversely, let point $(x_0,y_0)$ lie on the line $l_2$. Then, it satisfies the equation $(2)$, which means that $-4x_0+4y_0 = 12$. Dividing by $4$ we get $-x_0 + y_0 = 3,$ so the point $(x_0,y_0)$ satisfies the equation $(1)$ and therefore it lies on the line $l_1$. So, every point on $l_2$ also lies on $l_1$.

We conclude that the lines $l_1$ and $l_2$ consist of the same points, that is $l_1 = l_2$.

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So, what does multiplying of the equation by $4$ do to the line? Absolutely nothing. The "line speed" your teacher is talking about probably refers to parametrizations of the lines. For example, all points of the form $(t,t+3),\ t\in\mathbb R$, lie on the line $l_1$, and all the points that lie on $l_1$ are of that form. Of course, the same goes for $l_2$, since $l_2 = l_1$. Now, we can also look at all the points of the form $(4t,4t+3),\ t\in\mathbb R$. It turns out it's again the same set of points, but you can think of this second parametrization as "going faster" since if we think of $t$ as representing time, the first parametrization takes us from $(0,3)$ to $(1,4)$ in one unit time, while the second parametrization in one unit time takes us from $(0,3)$ to $(4,7)$, so the second parametrization describes "faster traversing of the same line".

However, these different parametrizations both satisfy both of the equations $(1)$ and $(2)$, and there is no canonical way to attach one parametrization to one equation and the other parametrization to the other equation, so in my opinion, multiplying by $4$ that occurs when manipulating the equations and multiplying by $4$ that occurs with the parametrizations don't have any connection with each other, besides using the fact that the function $t\mapsto 4t$ is a bijection on $\mathbb R$.