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Question
Find the equation of the line passing through the point of intersection of the lines 4x − 7y − 3 = 0 and 2x − 3y + 1 = 0 that has equal intercepts on the axes.
Solution
We have,
4x − 7y − 3 = 0 ... (1)
2x − 3y + 1 = 0 ... (2)
Solving (1) and (2) using cross-multiplication method:
\[\frac{x}{- 7 - 9} = \frac{y}{- 6 - 4} = \frac{1}{- 12 + 14}\]
\[ \Rightarrow x = - 8, y = - 5\]
Thus, the point of intersection of the given lines is \[\left( - 8, - 5 \right)\].
\[\therefore \frac{- 8}{a} - \frac{5}{a} = 1\]
\[ \Rightarrow - 8 - 5 = a\]
\[ \Rightarrow a = - 13\]
Hence, the equation of the required line is \[\frac{x}{- 13} + \frac{y}{- 13} = 1 \text { or } x + y + 13 = 0 .\]
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