Equation of a line (point/point of intersection)

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Find the equation of the line passsing through the point P $\left(-1,\frac{2}{5}\right)$ and the intersection of the lines $5x+5y-3= 0$ and $5x-9y-8= 0$

**Answer**

The lines $5x+5y-3= 0$ and $5x-9y-8= 0$ intersect at the point Q$\left(\frac{-67}{70},\frac{5}{14}\right)$

The given line passes through the point P $\left(-1,\frac{2}{5}\right)$ and $\left(\frac{-67}{70},\frac{5}{14}\right)$

The slope of the line joining the points $P\left(-1,\frac{2}{5}\right)$ and $Q\left(\frac{-67}{70},\frac{5}{14}\right)$ is given by $m=\frac{\frac{5}{14}-\frac{2}{5}}{\frac{-67}{70}+1}=-1$

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

$y =-x+\frac{2}{5}-\left(-1\right)\left(-1\right)$

or, $y =-x+\frac{2}{5}-1$

or, $y=-x-\frac{3}{5}$