Question

The simultaneous equations $2x+3y=5,4x+6y=10$, have

• A. no solution
• B. only one solution
• C. only two solutions
• D. several solutions

Answer 1

The correct option is D several solutions

We write the equations in the form $ax+by+c=0.$

Then, check for $\frac{a_1}{a_2}\ne \frac{b_1}{b_2},\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}and\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

The equations are

$2x+3y=5⟹2x+3y-5=0$

And, $4x+6y=10⟹4x+6y-10=0$

Here, $a_1=2,b_1=3,c_1=-5a_2=4,b_2=6,c_2=-10$

$\therefore \frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2},\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2},\frac{c_1}{c_2}=\frac{-5}{-10}=\frac{1}{2}$

$\therefore \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}.$

This is the condition for coincidence.