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Find the relation between \[x\] and \[y\] if the points \[A\left( {x,y} \right),B\left( { - 5,7} \right)\] and \[C\left( { - 4,5} \right)\] are collinear.

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Hint: We need to find the relation between \[x\] and \[y\] as all the given points are collinear. We know that the points \[A,B\] and \[C\] are collinear then area of \[\Delta ABC = 0\]. So, we will calculate area of triangle using \[A = \dfrac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]\] and put it equal to 0 to find the relation between \[x\] and \[y\].

Complete step by step solution: We will first consider the given points \[A\left( {x,y} \right),B\left( { - 5,7} \right)\] and \[C\left( { - 4,5} \right)\].
We need to find the relation between \[x\] and \[y\] if the given points are collinear.
Now, we know that the points \[A\left( {x,y} \right),B\left( { - 5,7} \right)\] and \[C\left( { - 4,5} \right)\] are collinear then area of \[\Delta ABC = 0\].
Thus, we will use \[A = \dfrac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]\] to find the area of triangle where \[\left( {{x_1},{y_1}} \right);\left( {{x_2},{y_2}} \right)\] and \[\left( {{x_3},{y_3}} \right)\] are the coordinated of \[A,B\] and \[C\].
Hence, we will substitute the value of coordinates given in the formula and find the value of area of triangle,
Thus, we get,
\[
   \Rightarrow A = \dfrac{1}{2}\left[ {x\left( {7 - 5} \right) - 5\left( {5 - y} \right) - 4\left( {y - 7} \right)} \right] \\
   \Rightarrow A = \dfrac{1}{2}\left[ {2x - 25 + 5y - 4y + 28} \right] \\
   \Rightarrow A = \dfrac{1}{2}\left[ {2x + y + 3} \right] \\
 \]
Now, as we know that the points \[A,B\] and \[C\] are collinear then area of \[\Delta ABC = 0\].
Thus, we get,
\[
   \Rightarrow \dfrac{1}{2}\left[ {2x + y + 3} \right] = 0 \\
   \Rightarrow 2x + y + 3 = 0 \\
   \Rightarrow 2x + y = - 3 \\
 \]

Hence, we get the relation between \[x\] and \[y\] is \[2x + y = - 3\].

Note: We must remember that the coordinates are collinear when the area of the triangle is equal to zero and using this fact only, we have found the relation between \[x\] and \[y\]. We have to remember the formula, \[A = \dfrac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]\] to determine the area of triangle using the coordinates of the triangle. We can easily determine the relation as one of the coordinates are given as \[\left( {x,y} \right)\].