Proof that if two lines are parallel then $A_1$ = $A_2$ and $B_1$ = $B_2$?

Question

Let two lines to be parallel in their general form.

$L_1$ : $A_1 x$ + $B_1 y$ + $C_1$

$L_2$ : $A_2 x$ + $B_2 y$ + $C_2$

Now i wish to prove $A_1$ = $A_2$ and $B_1$ = $B_2$

But i can only think of the prove in my head, not sure how to write it mathematically.

Answer 1

Your claim is wrong.

Write the formula of each line in the form $y=mx +c$

$L_1: y= (\frac{-A_1}{B_1})x + (\frac{-C_1}{B_1})$

$L_2: y= (\frac{-A_2}{B_2})x + (\frac{-C_2}{B_2})$

For two lines to parallel theirs slopes($m$) must be equal.

So, by comparison, it would be seen that $\frac{A_1}{B_1}=\frac{A_2}{B_2}$,

which doesn't necessarily $\Rightarrow A_1=A_2$ and $B_1=B_2$

Answer 2

Let $A_1, A_2, B_1,B_2 \not = 0$. Then

$$L_1 || L_2 \Leftrightarrow \frac {A_1}{A_2}=\frac {B_1}{B_2}$$ Proof:

Let $\frac {A_1}{A_2}\not=\frac {B_1}{B_2}$. Then the system of equations $$\begin{cases} A_1 x + B_1 y + C_1=0, \\ A_2 x + B_2 y + C_2=0 \end{cases} $$

has a solution. A point that belongs to both straight. The contradiction (because the lines are parallel)

Addition: For example:

$l_1: 2x+3y+7=0$ and $4x+6y+7=0$