Question

Without actually solving the simultaneous equations given below, decide whether simultaneous equations have unique solution, no solution or infinitely many solutions.

$\frac{x-2y}{3}=1;2x-4y=\frac{9}{2}$

• A. No solution
• B. Infinitely many solutions
• C. Unique solutions
• D. Data insufficient

Answer 1

The correct option is A No solution

The given equations are

$\frac{x-2y}{3}=1$ and $2x-4y=\frac{9}{2}$.

The equations can be written as,

$x-2y-3=0$ ..........(i)

and $4x-8y-9=0$ ........(ii)

We know that, for two linear equations

$a_{1}x+b_1=c_1$ and $a_{2}x+b_{2}y=c_2$:

(a) If $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$, the system has exactly one solution.

(b) If $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$, the system has infinitely many solutions.

(c) If $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$, the system has no solution.

Here $a_1=1,a_2=4,b_1=-2,b_2=-8,c_1=-3$ and $c_2=-9$

$\therefore \frac{a_1}{a_2}=\frac{1}{4},\frac{b_1}{b_2}=\frac{-2}{-8}=\frac{1}{4},\frac{c_1}{c_2}=\frac{-3}{-9}=\frac{1}{3}$

$\Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

Hence, the system of equations is inconsistent and has no solution.