Question

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

• A. Only one solution
• B. No solution
• C. Only two solutions
• D. Infinite number of solutions

Answer 1

Answer: (D)

Infinite number of solutions

Given equations: $2x-3y+5=0$ and $6y-4x=10$

Rewriting the equations, we get
$2x-3y+5=0$ ...$\left(i\right)$
$-4x+6y-10=0$ ...$\left(ii\right)$
Comparing the coefficients from $\left(i\right)$ and $\left(ii\right)$ gives
$\frac{a_1}{a_2}=\frac{2}{-4},\frac{b_1}{b_2}=\frac{-3}{6},\frac{c_1}{c_2}=\frac{5}{-10}$
i.e, $\frac{a_1}{a_2}=-\frac{1}{2},\frac{b_1}{b_2}=-\frac{1}{2},\frac{c_1}{c_2}=-\frac{1}{2}$
$\Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
Therefore, equations have infinite numbers of solutions.
Hence, Option $D$ is correct.