For what value of k does the system of equations $k x + 2 y = 5, 3 x - 4 y = 10$ have

(i) a unique solution,

(ii) no solution?

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Solution

Equations are written as
$k x + 2 y - 5 = 0 3 x - 4 y - 10 = 0 a_{1} = k, b_{1} = 2, c_{1} = - 5 a_{2} = 3, b_{2} = - 4, c_{3} = - 10$

(i) If $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ 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.

$\frac{k}{3} \neq \frac{2}{- 4} k \neq \frac{- 3}{2}$

(ii) The pair of linear equations has no solution if $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$

here,

$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \frac{k}{3} = \frac{2}{- 4} k = \frac{- 3}{2}$

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