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Find the centroid of the triangle whose sides are given by the following equations:

$$\begin{matrix} 4x & - & y & = &19 \\ x &- & y & = & 4 \\ x& + & 2y & = & -11 \end{matrix}$$

  1. $\left(\frac{11}{3}, -\frac{7}{3}\right)$
  2. $\left(\frac{5}{3}, -\frac{7}{3}\right)$
  3. $\left(-\frac{11}{3}, -\frac{7}{3}\right)$
  4. $\left(\frac{7}{3}, -\frac{11}{3}\right)$
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Given equations,

4x - y = 19----------------------@1

x - y = 4------------------------@2

x + 2y = -11 --------------------@3

solving 1st and 2nd, we'll get

$x_{1} = 5, y_{1} = 1$

solving 2nd and 3rd, we'll get 

$x_{2} = -1, y_{2} = -5$

solving 1st and 3rd, we'll get 

$x_{3} = 3, y_{3} = -7$

The 3 vertices of the triangle (5,1), (-1,-5) and (3,-7)

Centroid of the triangle,

$(O_{x}, O_{y}) = \left ( \frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3} \right ) = \left ( \frac{5 - 1 + 3}{3}, \frac{1 - 5 -7}{3}\right ) = \left ( \frac{7}{3},\frac{-11}{3} \right )$

Option D.

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