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The equations $2 x - 3 y + 5 = 0$ and $6 y - 4 x = 10$ , when solved simultaneously has:

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Only one solution

No solution

Only two solutions

Infinite number of solutions

answer is D.

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Detailed Solution

Given equations are,

2 x - 3 y + 5 = 0

and

6 y - 4 x = 10

.
Rewriting the equations, we get,

2 x - 3 y + 5 = 0

........(1)

- 4 x + 6 y - 10 = 0

.........(2)
Comparing the coefficients from

(1)

and

(2)

gives,

\frac{a_{1}}{a_{2}} = \frac{2}{- 4}, \frac{b_{1}}{b_{2}} = \frac{- 3}{6}, \frac{c_{1}}{c_{2}} = \frac{5}{- 10}

\Rightarrow \frac{a_{1}}{a_{2}} = - \frac{1}{2}, \frac{b_{1}}{b_{2}} = - \frac{1}{2}, \frac{c_{1}}{c_{2}} = - \frac{1}{2}

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

Therefore, equations have infinite numbers of solutions.
Therefore, option (4) is correct.

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