Equation of a line (point/point of intersection)

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

**Answer**

The lines $5x+y+9= 0$ and $4x+3y+4= 0$ intersect at the point Q$\left(\frac{23}{11},\frac{-16}{11}\right)$

The given line passes through the point P $\left(3,-1\right)$ and $\left(\frac{23}{11},\frac{-16}{11}\right)$

The slope of the line joining the points $P\left(3,-1\right)$ and $Q\left(\frac{23}{11},\frac{-16}{11}\right)$ is given by $m=\frac{\frac{-16}{11}+1}{\frac{23}{11}-3}=\frac{1}{2}$

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

$y =\frac{1}{2}x--\left(\frac{1}{2}\right)\left(3\right)$

or, $y =\frac{1}{2}x-1-\frac{3}{2}$

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