If 2x + 3y - 5z = 18, 3x + 2y + z = 29 and x + y + 3z = 17, then what is the value of xy + yz + zx ?

Solution(By Examveda Team)

$$\eqalign{ & 2x + 3y - 5z = 18\,......\left( 1 \right) \cr & 3x + 2y + z = 29\,......\left( 2 \right) \cr & x + y + 3z = 17\,......\left( 3 \right) \cr & {\text{Equation}}\left( 2 \right){\text{ and Equation}}\left( 1 \right),{\text{we get}} \cr & 3x + 2y + z = 29 \cr & \underline {2x + 3y - 5z = 18} \to \left( {{\text{Subtracting}}} \right) \cr & x - y + 6z = 11\,......\left( 4 \right) \cr & {\text{Adding Equation}}\left( 4 \right){\text{and Equation}}\left( 3 \right), \cr & 2x + 9z = 28\,......\left( 5 \right) \cr & {\text{And Equation}}\left( 2 \right){\text{and}}\,2 \times {\text{Equation}}\left( 3 \right), \cr & 3x + 2y + z = 29 \cr & \underline {2x + 2y + 6z = 34} \to \left( {{\text{Subtracting}}} \right) \cr & x - 5z = - 5\,......\left( 6 \right) \cr & {\text{Equation}}\left( 5 \right){\text{and}}\,2 \times {\text{Equation}}\left( 6 \right), \cr & 2x + 9z = 28 \cr & \underline {2x - 10z = - 10} \to \left( {{\text{Subtracting}}} \right) \cr & 19z = 38 \cr & \therefore z = 2 \cr & {\text{Now, from equation}}\left( {\text{6}} \right), \cr & x - 5 \times 2 = - 5 \cr & x = 10 - 5 \cr & x = 5 \cr & {\text{And from equation}}\left( {\text{3}} \right), \cr & 5 + y + 3 \times 2 = 17 \cr & y = 17 - 11 \cr & y = 6 \cr & \therefore xy + yz + zx \cr & = 5 \times 6 + 6 \times 2 + 5 \times 2 \cr & = 30 + 12 + 10 \cr & = 52 \cr} $$