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Question
Find the equation of the line passing through the intersection of the lines 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.
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Solution
The point of intersection of lines 2x + y = 5 and x + 3y + 8 = 0 is given by \[\left( \frac{23}{5}, - \frac{21}{5} \right)\]
Now, the slope of the line 3x + 4y = 7 \[\text { or } y = - \frac{3}{4}x + \frac{7}{4}\] is \[- \frac{3}{4}\]
Now, we know that the slopes of two parallel lines are equal.
So, the slope of the required line is \[- \frac{3}{4}\]
Now, the equation of the required line passing through \[\left( \frac{23}{5}, - \frac{21}{5} \right)\] and having slope \[- \frac{3}{4}\] is given by
\[y + \frac{21}{5} = - \frac{3}{4}\left( x - \frac{23}{5} \right)\]
\[ \Rightarrow y + \frac{21}{5} = - \frac{3}{4}x + \frac{69}{20}\]
\[ \Rightarrow y + \frac{3}{4}x = \frac{69}{20} - \frac{21}{5}\]
\[ \Rightarrow 20y + 15x = - 15\]
\[ \Rightarrow 3x + 4y + 3 = 0\]
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