fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 3)
$$? = 3x + 2y + z$$
$${\left( {2x + y - z} \right)^2} + {\left( {x - y} \right)^2} + {\left( {z - 3} \right)^2} = - {\left( {3y - z} \right)^2}\,\,\,\,\,\left( * \right)$$
$$\left. \matrix{
{\left( {2x + y - z} \right)^2} + {\left( {x - y} \right)^2} + {\left( {z - 3} \right)^2}\,\,\, \ge 0\,\, \hfill \cr
- {\left( {3y - z} \right)^2}\,\, \le 0 \hfill \cr} \right\}\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,{\left( {2x + y - z} \right)^2} + {\left( {x - y} \right)^2} + {\left( {z - 3} \right)^2} = 0\,\,\,\,\,\,\left[ { = - {{\left( {3y - z} \right)}^2}} \right]$$
$${\left( {2x + y - z} \right)^2} + {\left( {x - y} \right)^2} + {\left( {z - 3} \right)^2} = 0\,\,\,\,\,\,\, \Rightarrow \,\left\{ \matrix{
\,x - y = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,x = y\,\,\, \hfill \cr
\,z - 3 = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,z = 3\,\,\,\, \hfill \cr
\,2x + y - z = 0\,\,\,\,\mathop \Rightarrow \limits^{{\rm{both}}\,{\rm{above}}} \,\,\,\,\,3x - 3 = 0\, \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\left( {x,y,z} \right) = \left( {1,1,3} \right)$$
$$? = 3x + 2y + z = 3\left( 1 \right) + 2\left( 1 \right) + 3 = 8$$
The correct answer is (A).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
