Which of the following pairs of equations is not dependent?

3 x - 7 y = 10; 6 x - 14 = 20

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3 x - 7 y = 10; 6 x - 14 = 12

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4 x + 3 y = 5; 8 x + 6 y = 10

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7 x + 11 y = 10; 14 x + 22 y = 20

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Solution

The correct option is B $3 x - 7 y = 10; 6 x - 14 = 12$
Consider a pair of linear equations :
$a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$
The equations are said to be dependent if $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ ...(i)
Consider the equations $3 x - 7 y = 10 a n d 6 x - 14 = 12.$
Here, $a_{1} = 3, b_{1} = - 7 a n d c_{1} = 10$
Similarly, $a_{2} = 6, b_{2} = - 14 a n d c_{2} = 12$
Substituting these value in eq. (i), we get
$\frac{3}{6} = \frac{- 7}{- 14} \neq \frac{10}{12}$
Therefore, the equations $x + 6 x = 7$ are not dependent.

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