# Verify if the lines are parallel or coincidental? 7x+3y+1=0 21x+9y-5=0Verify if the lines are parallel or coincidental? 7x+3y+1=0 21x+9y-5=0

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First we need to calculate the slopes for both lines:

If slopes are equal, then the lines are parallel.

7x + 3y + 1 = 0

==> 3y = -7x - 1

==> y= -(7/3)x - 1/3

==> m1 = -7/3

21x + 9y - 5 = 0

==> 9y = -21x + 5

==> y= -21/9 x + 5/9

==> y= -7/3 x + 5/9

==> m2= -7/3

We notice that:

m1= m2

==> both lines are parallel.

Parallel lines have the same slope.

The first thing you need to do is to determine the slope of each line. To do this, you need to write the equations in the form y = mx + b, in which m = slope.

7x + 3y + 1 = 0

3y = -7x -1 subtract 7x & 1 from each side of the equation

y = -7/3x -1/3 divide both sides of the equation by 3

m= -7/3 determine the slope

21x + 9y - 5 =0

9y = -21x +5 subtract 21x & add 5 to each sides of the eqtn

y = -21/9x + 5/9 divide both sides of the equation by 9

m = -21/9 = -7/3 determine the slopes

Both equations have a slope of -7/3, so the two lines are parallel

The equation of a line can be expressed in the form y=mx+b, where the slope of the line is m and b is the y-intercept.

Parallel lines have the same slope.

For the given lines:

7x+3y+1=0

=>3y=-7x-1

=>y=(-7/3)x-(1/3)

Therefore the slope is -7/3

21x+9y-5=0

=>9y=-21x+5

=>y=(-21/9)x+(5/9)

Here the slope is -21/9=-7/3

**Therefore as the two lines have an equal slope they are parallel.**

7x+3y +1 = 0

21x+9y-5 = 0

Two lines a1x+b1y+c1= 0 and a2x+b2y+c2 = 0 are said to be coinciding if we acan find a constant k, for which,

(a1x+b1x+c2)k - a2x+b2y+c2 vanishes . Or

a1/a2 = b1/b2= c1/c2 = k.

Here, a1 = 7, b1 = 3 and c1 = 1

a2 = 21, a2 = 9 c2 = -5.

Therefore a1/a2 = b1/b2 = 1/3. But c1/c2 = -1/6.

So the lines given are not coincident.

They are || as their slope -b1/a1 = -3/7 and -b2/a2 = -9/21 = -3/7 are same.

If the 2 lines are parallel, then:

a1/a2 = b1/b2

If the 2 lines are coincidental:

a1/a2 = b1/b2 = c1/c2

These constraints are applied when the equations of the lines are put in general form.

We have 2 lines:

d1: a1x+b1y+c1=0

d2: a2x+b2y+c2=0

We'll identify a1, a2, b1, b2, c1, c2:

a1=7 and a2 = 21

b1 = 3 and b2 = 9

c1 = 1 and c2 = -5

Now, we'll compose the ratios:

a1/a2 = 7/21

We'll divide by 7:

a1/a2 = 1/3

b1/b2 = 3/9

We'll divide by 3:

b1/b2 = 1/3

c1/c2 = 1/-5

c1/c2 = -1/5

c1/c2 = -0.2

It is obvious that a1/a2 = b1/b2 = 1/3 but they are not equal to c1/c2 = -0.2.

**So, the lines d1 and d2 are parallel, but not coincidental.**