# How to prove if 2 lines are parallel or coincidental? d1: 5x+3y-4=0 d2:10x+6y-7=0

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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:

d1: a1x+b1y+c1=0

d2: a2x+b2y+c2=0

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.