Question

For what value of k ,the first pair of linear equations kx-4y=3,6x-12y=9 has infinite number of solution

Answer 1

Answer :

k = 2

Note:

★ A linear equation is two variables represent a straight line .

★ The word consistent is used for the system of equations which consists any solution .

★ The word inconsistent is used for the system of equations which doesn't consists any solution .

★ Solution of a system of equations : It refers to the possibile values of the variable which satisfy all the equations in the given system .

★ A pair of linear equations are said to be consistent if their graph ( Straight line ) either intersect or coincide each other .

★ A pair of linear equations are said to be inconsistent if their graph ( Straight line ) are parallel .

★ If we consider equations of two straight line

ax + by + c = 0 and a'x + b'y + c' = 0 , then ;

• The lines are intersecting if a/a' ≠ b/b' .

→ In this case , unique solution is found .

• The lines are coincident if a/a' = b/b' = c/c' .

→ In this case , infinitely many solutions are found .

• The lines are parallel if a/a' = b/b' ≠ c/c' .

→ In this case , no solution is found .

Solution :

Here ,

The given linear equations are ;

kx - 4y = 3 => kx - 4y - 3 = 0

6x - 12y = 9 => 6x - 12y - 9 = 0

Clearly , we have ;

a = k

a' = 6

b = -4

b' = -12

c = -3

c' = -9

Now ,

For the given equations to have infinite number of solutions , we have ;

a/a' = b/b' = c/c'

Thus ,

=> k/6 = -4/-12 = -3/-9

=> k/6 = 1/3 = 1/3

=> k/6 = 1/3

=> k = ⅓ × 6

=> k = 6/3

=> k = 2

Hence , k = 2 .

Answer 2

$\huge \blue {Answer}$

Here, we have,
kx - 4y = 3 and 6x - 12y = 9

We have,

${a_{1} } = k, {b_{1} } = -4, {c_{1} } = 3$

${a_{2} } = 6, {b_{2} } = -12, {c_{2} } = 9$

We know that in condition of infinitely many solutions:-

$\frac{a_{1} }{a_{2} } = \frac{b_{1} }{b_{2} } = \frac{c_{1} }{c_{2} }$

$\frac{k}{6} = \frac{ - 4}{ - 12} = \frac{3 }{9}$

$\frac{k}{6} = \frac{ 1}{ 3} = \frac{1 }{3}$

$\frac{k}{6} = \frac{1}{3}$

$3k = 6 \times 1$

$3k = 6$

$k = \frac{6}{3}$

$\boxed {k=2}$

Therefore, given system of equation have infinitely many solutions when value of k = 2.