Equation of a line (point/point of intersection)

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Find the equation of the line passsing through the point P $\left(\frac{7}{6},\frac{-1}{8}\right)$ and the intersection of the lines $4x-y+9= 0$ and $9x-8y+6= 0$

**Answer**

The lines $4x-y+9= 0$ and $9x-8y+6= 0$ intersect at the point Q$\left(\frac{66}{23},\frac{57}{23}\right)$

The given line passes through the point P $\left(\frac{7}{6},\frac{-1}{8}\right)$ and $\left(\frac{66}{23},\frac{57}{23}\right)$

The slope of the line joining the points $P\left(\frac{7}{6},\frac{-1}{8}\right)$ and $Q\left(\frac{66}{23},\frac{57}{23}\right)$ is given by $m=\frac{\frac{57}{23}+\frac{1}{8}}{\frac{66}{23}-\frac{7}{6}}=\frac{1437}{940}$

The equation of the line is$$y - y_0 = m\left(x - x_0\right)$$

or, $$y= mx + y_0-mx_0$$

Substituting the values of $m$, $x_0$ and $y_0$, we get

$y =\frac{1437}{940}x-\frac{1}{8}-\left(\frac{1437}{940}\right)\left(\frac{7}{6}\right)$

or, $y =\frac{1437}{940}x-\frac{1}{8}-\frac{3353}{1880}$

or, $y=\frac{1437}{940}x-\frac{897}{470}$