Answer
a)The center of mass is$\left[\begin{array}{ c }10/3\\2\end{array}\right]$
b) Add 3.5 g at (0, 1), add .5 g at (8, 1), and add 2 g at (2, 4).
Work Step by Step
a)The center of mass is $\frac{1}{3}(1\cdot \left[\begin{array}{ l }0\\1\end{array}\right]+1\cdot \left[\begin{array}{ l }8\\1\end{array}\right]+1\cdot \left[\begin{array}{ l }2\\4\end{array}\right])=\left[\begin{array}{ c }10/3\\2\end{array}\right]$
b) The total mass of the new system is 9 grams. The three masses added, w1, w2, and w3, satisfy the equation
$\frac{1}{3}(1\cdot \left[\begin{array}{ l }0\\1\end{array}\right]+1\cdot \left[\begin{array}{ l }8\\1\end{array}\right]+1\cdot \left[\begin{array}{ l }2\\4\end{array}\right])=\left[\begin{array}{ c }10/3\\2\end{array}\right]$
which can be rearranged to
$\frac{1}{9}((w_1+1)\cdot \left[\begin{array}{ l }0\\1\end{array}\right]+(w_2+1)\cdot \left[\begin{array}{ l }8\\1\end{array}\right]+(w_3+1)\cdot \left[\begin{array}{ l }2\\4\end{array}\right])=\left[\begin{array}{ l }2\\2\end{array}\right]$
and
$(w_1+1)\cdot \left[\begin{array}{ l }0\\1\end{array}\right]+(w_2+1)\cdot \left[\begin{array}{ l }8\\1\end{array}\right]+(w_3+1)\cdot \left[\begin{array}{ l }2\\4\end{array}\right]=\left[\begin{array}{ c }18\\18\end{array}\right]$
The condition w1 + w2 + w3 = 6 and the vector equation above combine to produce a system of three equations whose augmented matrix is shown below, along with a sequence of row operations:
$\left[\begin{array}{ r r r r }1&1&1&6\\0&8&2&8\\1&1&4&12\end{array}\right]\sim \left[\begin{array}{ l l l l }1&1&1&6\\0&8&2&8\\0&0&3&6\end{array}\right]\sim \left[\begin{array}{ c c c c }1&1&1&6\\0&8&2&8\\0&0&1&2\end{array}\right]$
$\sim \left[\begin{array}{ l l l l }1&1&0&4\\0&8&0&4\\0&0&1&2\end{array}\right]\sim \left[\begin{array}{ l l l r }1&0&0&3.5\\0&8&0&4\\0&0&1&2\end{array}\right]\sim \left[\begin{array}{ c c c c }1&0&0&3.5\\0&1&0&.5\\0&0&1&2\end{array}\right]$
Add 3.5 g at (0, 1), add .5 g at (8, 1), and add 2 g at (2, 4).
