Equation of a line (point/point of intersection)

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Find the equation of the line passsing through the point P $\left(\frac{4}{9},\frac{3}{7}\right)$ and the intersection of the lines $6x+5y-1= 0$ and $9x-9y-10= 0$

**Answer**

The lines $6x+5y-1= 0$ and $9x-9y-10= 0$ intersect at the point Q$\left(\frac{-59}{99},\frac{17}{33}\right)$

The given line passes through the point P $\left(\frac{4}{9},\frac{3}{7}\right)$ and $\left(\frac{-59}{99},\frac{17}{33}\right)$

The slope of the line joining the points $P\left(\frac{4}{9},\frac{3}{7}\right)$ and $Q\left(\frac{-59}{99},\frac{17}{33}\right)$ is given by $m=\frac{\frac{17}{33}-\frac{3}{7}}{\frac{-59}{99}-\frac{4}{9}}=\frac{-60}{721}$

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

$y =-\frac{60}{721}x+\frac{3}{7}-\left(-\frac{60}{721}\right)\left(\frac{4}{9}\right)$

or, $y =-\frac{60}{721}x+\frac{3}{7}+\frac{80}{2163}$

or, $y=-\frac{60}{721}x+\frac{1007}{2163}$