Lines parallel to a given line

Question

I have to show that any line parallel to $Ax+ By + c =0$ is of the form $Ax + By + k =0$

How do I show this?

Thank you!

Answer 1

(Based on zar's answer and Moron's comment there.)

If two lines $Ax+By+C=0$ and $Dx+Ey+F=0$ are parallel, then they have no points of intersection (or infinitely many if you allow a line to be parallel to itself). In either case, the system of equations $$\begin{cases} Ax+By+C=0,\\\\ Dx+Ey+F=0 \end{cases}$$ will not have a single, unique solution. The system of equations is equivalent to $$\begin{bmatrix} A & B\\\\ D & E \end{bmatrix}\cdot\begin{bmatrix} x\\\\ y \end{bmatrix}+\begin{bmatrix} C\\\\ F \end{bmatrix}=0$$ or $$\begin{bmatrix} A & B\\\\ D & E \end{bmatrix}\cdot\begin{bmatrix} x\\\\ y \end{bmatrix}=\begin{bmatrix} -C\\\\ -F \end{bmatrix}.$$ This matrix equation has a single unique solution if and only if the coefficient matrix $\begin{bmatrix} A & B\\\\ D & E \end{bmatrix}$ is invertible, which is true if and only if its determinant is nonzero.

So, if the lines are parallel, then $$\begin{vmatrix} A & B\\\\ D & E \end{vmatrix}=AE-BD=0$$ or $$\frac{A}{D}=\frac{B}{E}$$ which means that $\frac{A}{D}(Dx+Ey+F)=Ax+By+\frac{FA}{D}=0$, so $Dx+Ey+F=0$ can be written in the form $Ax+By+k=0$.

edit: from Arturo Magidin's comment below, to handle the cases where $D=0$ or $E=0$ (in which $A/D = B/E$ makes no sense):

If $D=0$ then $A=0$ or $E=0$, but you cannot have both $D$ and $E$ zero. So $D=0$ implies $A=0$, and $B\neq 0\neq E$. Similarly, if $E=0$, then $B=0$ and $A\neq 0 \neq D$. And you get the similar conclusion without forgetting about horizontal and vertical lines.

Answer 2

Examples don't show it in any case. You cannot prove that something holds for infinitely many things by doing nothing except checking finitely many examples.

What you need to do is remember that two lines are parallel if and only if they have the same slope or they are both vertical (of the form $x=k$ for some constant $k$). Then figure out what the slope of an arbitrary line $\alpha x+\beta y+\gamma=0$ would be in terms of $\alpha$, $\beta$, and $\gamma$ (careful when dividing by constants to consider what happens if the constant you are dividing by is equal to zero). Then see what this means if you want to find all lines $\alpha x+\beta y+\gamma = 0$ that are parallel to a given line $Ax+By+c=0$.

Answer 3

You can solve the following system of equations:

$$ \begin{cases} Ax+By+c=0,\\ Ax+By+k=0
\end{cases} $$

If you subtract the second line from the firs, you obtain $c-k=0$. Then, if $c\ne k$, you have no solution: the lines are parallel.

Answer 4

It is a preliminary and useful exercise to show that the line $Ax+By+C=0$ and the vector $v=(A,B)$ are perpendicular (one says that $v$ is the normal vector of the line). With these, all amounts to observe that two lines are parallel if and only if their normal vectors are parallel.