Question

pairs of linear equations intersect at a point / are parallel / are coincident . a) 5x-4y+8=0 7x+6y-9=0​

Answer 1

$\huge\underline\bold\orange{Answer:-}$

$\large\underline\bold\green{Given:-}$

$\implies$pair of equations 5x – 4y + 8 =0 and 7x + 6y -9 = 0

$\large\underline\bold\red{To\:Prove:-}$

$\implies$The lines representing the pair  are parallel , coincident , Intersect at a point or none of these

$\large\underline\bold\red{solution:-}$

Two lines

$\implies$ a₁x  + b₁y  + c₁ = 0

$\implies$ a₂x  + b₂y  + c₂ = 0

parallel  if  a₁/a₂  = b₁/b₂  

Coincident if a₁/a₂  = b₁/b₂  =  c₁ / c₂

Intersect at a point  if a₁/a₂ ≠ b₁/b₂  

Lets check    for

$\dashrightarrow$5x – 4y + 8 =0

$\dashrightarrow$7x + 6y -9 = 0

$\dashrightarrow$a₁/a₂  = 5/7    

$\dashrightarrow$b₁/b₂  =  - 4/6  

$\dashrightarrow$c₁ / c₂  = 8/-9

$\dashrightarrow$5/7      ≠  - 4/6  

$\dashrightarrow$a₁/a₂   ≠ b₁/b₂  

$\small\underline\bold\red{lines\:intersect\:at\:a\:point}$

Answer 2

Given: a pair of equations 5x -4y+8=0 and 7x+6y-9=0

To find: the lines representing the pair are parallel coincident, intersect at a point or none of these.

Solution :

Two lines

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

parallel if a₁/a₂ = b₁/b₂

Coincident if a₁/a₂ = b₁/b₂ = c₁ / c₂

Intersect at a point if a₁/a₂ ≠ b₁/b₂

Lets check for

5x – 4y + 8 =0

7x + 6y -9 = 0

a₁/a₂ = 5/7

b₁/b₂ = - 4/6

c₁ / c₂ = 8/-9

5/7 ≠ - 4/6

a₁/a₂ ≠ b₁/b₂

=> lines intersect at a point