How to prove if 2 lines are parallel or coincidental? d1: 5x+3y-4=0 d2:10x+6y-7=0
d1: 5x + 3y - 4 = 0
d2: 10 x + 6y - 7 = 0
Let us find the ratio d2/d1:
If d1 and d2 are coincidental, then:
a2/a1 = b2/b1 = c2/c1
10/5 = 6/3 = -7/-4
2 = 2 = 7/4
Then d1 and d2 are not coincidental.
Now let us determine if d1 and d2 are parallel.
First, let us calculat the slopes m1 and m2
for d1, m1 = -5/3
for d2 , m2 = -10/6 = -5/3
m1 = m2
Then d1 and d2 are parallel bu NOT coincidental.
First, we notice that the equations of the lines are written in general form.
We have 2 lines:
If the 2 lines are parallel, then:
a1/a2 = b1/b2
If the 2 lines are coincidental:
a1/a2 = b1/b2 = c1/c2
We'll identify a1, a2, b1, b2, c1, c2:
a1=5 and a2 = 10
b1 = 3 and b2 = 6
c1 = -4 and c2 = -7
Now, we'll compose the ratios:
a1/a2 = 5/10
We'll divide by 5:
a1/a2 = 1/2
b1/b2 = 3/6
We'll divide by 3:
b1/b2 = 1/2
c1/c2 = -4/-7
c1/c2 = 4/7
It is obvious that a1/a2 = b1/b2 = 1/2 but they are not equal to c1/c2 = 4/7.
So, the lines d1 and d2 are parallel and not coincidental.