Equation of a line (point/point of intersection)
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Question

Find the equation of the line passsing through the point P $\left(0,\frac{5}{3}\right)$ and the intersection of the lines $5x+4y-5= 0$ and $3x+8y+1= 0$

Answer

The lines $5x+4y-5= 0$ and $3x+8y+1= 0$ intersect at the point Q$\left(\frac{-11}{7},\frac{5}{7}\right)$

The given line passes through the point P $\left(0,\frac{5}{3}\right)$ and $\left(\frac{-11}{7},\frac{5}{7}\right)$

The slope of the line joining the points $P\left(0,\frac{5}{3}\right)$ and $Q\left(\frac{-11}{7},\frac{5}{7}\right)$ is given by $m=\frac{\frac{5}{7}-\frac{5}{3}}{\frac{-11}{7}}=\frac{20}{33}$

The $y-intercept = \frac{5}{3}$.

So, the equation of the line is

$y=\frac{20}{33}x+\frac{5}{3}$