Find the relation between $x$ and $y$ if the points $ A( x,\ y) ,\ B( -5,\ 7)$ and $\displaystyle C( -4,\ 5)$ are collinear.

Given: Points $A\left( x,\ y\right) ,\ B\left( -5,\ 7\right)$ and $C( -4,\ 5)$ are collinear.

To do: To find the relation between x and y.

Solution:

Given, the points $A( x,\ y),\ B( -5,\ 7)$ and $C( -4,\ 5)$ are collinear.

So, the area formed by these vertices $A( x,\ y),\ B( -5,\  7)$ and $C( -4,\  5)$ will be zero.

And we know that area of a triangle with vertices $(x_{1} ,\ y_{1}),\ (x$_{2}$, y_{2})$ and (x$_{3}$, y$_{3}$), the area formed by the vertices,

A=$\frac{1}{2}\left[ x_{1}\left( y_{2} -y_{3}\right) +x_{2}\left( y_{3} -y_{1}\right) +x_{3}\left( y_{1} -y_{2}\right)\right]$

By using the formula, we have

$0=\frac{1}{2}\left[ x\left( 7-5\right) -5\left( 5-y\right) -4\left( y-7\right)\right]$

$\Rightarrow 2x-25+5y-4y+28=0$

$\Rightarrow 2x+y+3=0$

$\Rightarrow y=-2x-3$

Therefore, The relationship between $x$ and $y$ is,

$y=-2x-3$.