Question

which type of linear equations 2x-3y=4 and 6x-9y=12 represent​

Answer 1

The given linear equations are:

2x - 3y = 4 and 6x - 9y = 12

Here, $a_{1}$ = 2, $b_{1}$ = - 3, $c_{1}$ = 4 and  $a_{2}$ = 6, $b_{2}$ = - 9, $c_{2}$ = 12

We have to find, the given linear equations is.

Solution:

$\dfrac{a_{1} }{a_{2} }$ = $\dfrac{2 }{6 }=\dfrac{1 }{3 }$,

$\dfrac{b_{1} }{b_{2} }$ = $\dfrac{-3 }{-9 }=\dfrac{1 }{3 }$ and

$\dfrac{c_{1} }{c_{2} }$ = $\dfrac{4}{12}=\dfrac{1 }{3}$

∵  $\dfrac{a_{1} }{a_{2} }$ = $\dfrac{b_{1} }{b_{2} }$ = $\dfrac{c_{1} }{c_{2} }$ = $\dfrac{1}{3}$ has an infinite number of solutions.

Thus, the given linear equations has infinite number of solutions.

Answer 2

Given :   2x-3y=4

6x-9y=12  

To Find  : which type of linear equations 2x-3y=4 and 6x-9y=12 represent​

Solution:

a₁x +b₁y +c₁=0

a₂x +b₂y +c₂=0

a₁/a₂ ≠  b₁/b₂

then System is called consistent and have a unique Solution

Lines intersect each other at unique point

a₁/a₂ =  b₁/b₂ = c₁/c₂

then Consistent ( infinite solution )  as both Equation represent same line

a₁/a₂ =  b₁/b₂ ≠ c₁/c₂

Inconsistent ( no Solution )

as Lines are parallel and never intersect each other

2x-3y=4

6x-9y=12  

a₁ = 2     b₁ = - 3    c₁ = 4

a₂ = 6    b₂ = -9    c₂ = 12

a₁ /a₂  = 2/6 = 1/3

b₁/b₂ = -3/-9 = 1/3

c₁/c₂ = 4/12 = 1/3

a₁/a₂ =  b₁/b₂ = c₁/c₂

Hence ( infinite solution )  as both Equation represent same line

Both equations are of same line

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