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Without actually solving the simultaneous equations given below, decide whether the system has unique solution, no solution or infinitely many solutions. $3 y = 2 - x; 3 x = 6 - 9 y$ .

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Infinite solutions

Unique solution

No solution

Cannot be determined

answer is A.

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Detailed Solution

Given that,

3 y = 2 - x; 3 x = 6 - 9 y

.
As we know the condition for the pair of equations

a_{1} x + b_{1} y + c_{1} = 0

and

a_{2} x + b_{2} y + c_{2} = 0

to have infinite solutions is

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

.
From given equations,

a_{1} = 1, a_{2} = 3, b_{1} = 3, b_{2} = 9, c_{1} = - 2

and

c_{2} = - 6

.
We have

\frac{a_{1}}{a_{2}} = \frac{1}{3}

.
Also,

\frac{b_{1}}{b_{2}} = \frac{3}{9}

\Rightarrow \frac{b_{1}}{b_{2}} = \frac{1}{3}

And,

\frac{c_{1}}{c_{2}} = \frac{- 2}{- 6}

\Rightarrow \frac{c_{1}}{c_{2}} = \frac{1}{3}

Then,

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

\Rightarrow

13 = 39 = − 2 − 6

All the ratios are equal.
Thus, the system has infinite number of solutions.
Therefore, option (1) is correct.

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