Question 2 (ii)
On comparing the ratios $\frac{a_{1}}{a_{2}}$ , $\frac{b_{1}}{b_{2}}$ and $\frac{c_{1}}{c_{2}}$ , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.
9x + 3y + 12 = 0
18x + 6y + 24 = 0

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Solution

9x + 3y + 12 = 0
18x + 6y + 24 = 0
Comparing these equations with $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ , we get
$a_{1} = 9, b_{1} = 3, a n d c_{1} = 12$
$a_{2} = 18, b_{2} = 6 a n d c_{2} = 24$

$\frac{a_{1}}{a_{2}} = \frac{9}{18} = \frac{1}{2}$

$\frac{b_{1}}{b_{2}} = \frac{3}{6} = \frac{1}{2}$ and

$\frac{c_{1}}{c_{2}} = \frac{12}{24} = \frac{1}{2}$

Hence, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$

Therefore, the lines representing the given pair of linear equations are coincident.

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On comparing the ratios $\frac{a_{1}}{a_{2}}$ , $\frac{b_{1}}{b_{2}}$ and $\frac{c_{1}}{c_{2}}$ , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.
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