$6 x - 3 y + 10 = 0 and 2 x - y + 9 = 0$ has/have ____.

unique solution

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infinite solutions

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no solution

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can't be determined

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Solution

The correct option is C no solution
Given: $a_{1} = 6, b_{1} = - 3, c_{1} = 10 and a_{2} = 2, b_{2} = - 1, c_{2} = 9$
We know that when two equations are represented graphically, if the lines intersect at one point then, the pair of linear equations has a unique solution.
If the lines are parallel to each other then, the pair of equations do not have any solutions.
If the lines are coincident then, the pair of linear equations have infinitley many solutions.
1) For intersecting lines, $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} .$
Here, $\frac{a_{1}}{a_{2}} = \frac{6}{- 2}, \frac{b_{1}}{b_{2}} = \frac{- 3}{- 1} \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = 3$
Hence, the lines are not intersecting.

2) For parallel lines $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} .$
$\frac{6}{- 2} = \frac{- 3}{- 1} \neq \frac{10}{9} 3 = 3 \neq \frac{10}{9}$
Thus, the given pair of equations have no solution.

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