Question

By comparing the ratios a1/a2, b1/b2, c1/c2 find out whether the lines represented by the following pairs of linear equations intersect at a point, are parallel or are coincident.a) 5x- 4y + 8 = 0 7x+6y-9 = 0
b) 9x+3y + 12 = 0 18x+6y + 24 = 0
c) 6x - 3y + 10 = 0 2x - y + 9 = 0

Answer 1

SOLUTION :  

Given :  

a) 5x- 4y + 8 = 0 & 7x+ 6y - 9 = 0

b) 9x + 3y + 12 = 0 & 18x + 6y + 24 = 0

c) 6x - 3y + 10 = 0 & 2x - y + 9 = 0

(a)

On comparing with  a1x + b1y +c1 = 0 &   a2x + b2y + c2 = 0

a1= 5 , b1= - 4 , c1= 8

a2= 7, b2= 6 , c2 = -9

Now,

a1/a2 = 5/7 ,  b1/b2 = - 4/6,  c1/c2= 8/-9

Since, a1/a2 ≠ b1/b2

Hence, the lines representing the pair of linear equations are INTERSECTING at a point and have exactly one solution.

(b) 9x + 3y + 12 = 0 & 18x + 6y + 24 = 0

On comparing with  a1x + b1y +c1 = 0 &  a2x + b2y + c2 = 0

a1= 9 , b1= 3, c1= 12

a2= 18, b2= 6 , c2 = 24

Now,

a1/a2 = 9/18= 1/2 ,  b1/b2 = 3/6= 1/2 , c1/c2= 12/24= 1/2

Since, a1/a2 = b1/b2=c1/c2

Hence, the lines representing the pair of linear equations are COINCIDENT LINES and have infinitely many solutions.

c) 6x - 3y + 10 = 0 & 2x - y + 9 = 0

On comparing with  a1x + b1y +c1 = 0 &  a2x + b2y + c2 = 0

a1= 6 , b1= -3, c1= 10

a2=2, b2= -1, c2 = 9

Now,

a1/a2 = 6/2 ,  b1/b2 = -3/-1= 3, c1/c2= 10/9

Since, a1/a2 = b1/b2 ≠ c1/c2

Hence, the lines representing the pair of linear equations are PARALLEL LINES and have no many solution.

HOPE THIS ANSWER WILL HELP YOU

Answer 2

Hi ,

a ) Given :

5x - 4y + 8 = 0 ;

7x + 6y - 9 = 0

We have a1 = 5 , b1 = -4 , c1 = 8 ,

a2 = 7 , b2 = 6 , c2 = -9

Now , a1/a2 = 5/7 ; b1/b2 = -4/6 = -2/3

Therefore ,

a1/a2 ≠ b1/b2

So the given pair of linear equations

are intersecting lines and have unique

solution.

b ) Given :

9x + 3y + 12 = 0; 18x + 6y + 24 = 0

Comparing the given equations with

a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0

We have a1 = 9 , b1 = 3 , c1 = 12 ;

a2 = 18 , b2 = 6 , c2 = 24 ;

Now ,

a1/a2 = 9/18 = 1/2 ;

b1/b2 = 3/6 = 1/2 ;

c1/c2 = 12/24 = 1/2 ;

a1/a2 = b1/b2 = c1/c2 = 1/2

So, the given pair of linear equations

are coincident and have infinite number

of solution.

c ) Given :

6x - 3y + 10 = 0 ; 2x - y + 9 = 0

Comparing the given equations with

a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0

We have a1 = 6 , b1 = -3 , c1 = 10 ;

a2 = 2 , b2 = -1 , c2 = 9

Now ,

a1/a2 = 6/2 = 3 ;

b1/b2 = -3/( - 1 ) = 3 ;

c1/c2 = 10/9

Therefore ,

a1/a2 = b1/b2 ≠ c1/c2

So , the lines representing the pair of

linear equations are parallel and have

no solution.

I hope this helps you.

: )