MathematicsThe pair of equations 3x+4y=k, 9x+12y=6 has infinitely many solutions if

The pair of equations $3 x + 4 y = k, 9 x + 12 y = 6$ has infinitely many solutions if

A
$k = 2$
B
$k = 6$
C
$k \neq 6$
D
$k = 3$

Solution:

The given equations are,

3 x + 4 y - k = 0

9 x + 12 y - 6 = 0

Here,

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

,
And,

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

.
For the system to have infinite solutions condition is

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

Here,

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

⇒

\frac{4}{12} = \frac{- k}{- 6}

\Rightarrow \frac{1}{3} = \frac{- k}{- 6}

⇒

3 k = 6

⇒

k = 2

Hence, the correct option is 1.

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