For what value of k do the equations kx-2y=3 and 3x+y=5 represent two lines intersecting at a unique point?

(a) k=3 (b) k=-3

(c) k=6 (d) all real values except -6

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Solution

IIf $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ then the pair of linear equations
$a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$ has a unique solution.

Given pair of equations
$k x - 2 y = 3$
$k x - 2 y - 3 = 0$
and
$3 x + y = 5$
$3 x + y - 5 = 0$

Here $a_{1} = k, b_{1} = - 2, a_{2} = 3, b_{2} = 1$ .

$\frac{k}{3} \neq \frac{- 2}{1}$

$k \neq - 6$

(d) $k \neq - 6$

(d) all real values except -6

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