Determine how many solutions exist for the following pair of equations $8 x + 5 y = 9, 16 x + 10 y = 27$ .

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Both equations form coincident line hence they have no solutions.

Both equations form parallel lines hence they have no solutions.

Both equations form intersecting lines hence they have no solutions.

None of these

answer is B.

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Detailed Solution

Given equations are,

8 x + 5 y = 9, 16 x + 10 y = 27

.
The following pair of equations can be written as:

8 x + 5 y - 9 = 0; 16 x + 10 y - 27 = 0;

Comparing with the general equations

a_{1} x + b_{1} y + c_{1} = 0; a_{2} x + b_{2} y + c_{2} = 0

\Rightarrow a_{1} = 8, b_{1} = 5, c_{1} = - 9, a_{2} = 16, b_{2} = 10, c_{2} = - 27

Here,

\frac{a_{1}}{a_{2}} = \frac{8}{16} = \frac{1}{2}

\frac{b_{1}}{b_{2}} = \frac{5}{10} = \frac{1}{2}

\frac{c_{1}}{c_{2}} = \frac{- 9}{- 27} = \frac{1}{3}

,
Now,

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

this means the lines are parallel to each other.
We can see

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

.
Hence, the given pair of the linear equation is inconsistent and has no solution.
Therefore, option (2) is correct.

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