Chapter 9 - Systems and Matrices - 9.1 Systems of Linear Equations - 9.1 Exercises - Page 857: 30

$$\left\{ {\left( {66, - 34} \right)} \right\}$$

$$\eqalign{ & \frac{{x + 6}}{5} + \,\,\frac{{2y - x}}{{10}}\, = 1\,\,\,\,\,\,\,\, \cr & \,\,\frac{{x + 2}}{4} + \frac{{3y + 2}}{5}\, = - 3\,\,\, \cr & \cr & {\text{Clear the denominators}} \cr & \,10\left( {\frac{{x + 6}}{5}} \right) + \,\,10\left( {\frac{{2y - x}}{{10}}} \right)\, = 10\left( 1 \right)\,\,\,\,\,\,\,\, \cr & \,20\left( {\frac{{x + 2}}{4}} \right) + 20\left( {\frac{{3y + 2}}{5}} \right)\, = 20\left( { - 3} \right)\,\,\, \cr & \cr & 2\left( {x + 6} \right) + \,\left( {2y - x} \right)\, = 10\,\,\,\, \cr & 5\left( {x + 2} \right) + 4\left( {3y + 2} \right)\, = - 60 \cr & \cr & 2x + 12 + \,2y - x\, = 10\,\,\,\, \cr & 5x + 10 + 12y + 8\, = - 60 \cr & \cr & \,\,\,\,x + \,2y = - 2\,\,\,\,\,\,\left( {\bf{1}} \right) \cr & 5x + 12y = - 78\,\,\,\left( {\bf{2}} \right) \cr & \cr & {\text{Multiply the equation }}\left( {\bf{1}} \right){\text{ by }} - 5 \cr & - 5\,x - 10y = 10\,\,\,\,\, \cr & 5x + 12y = - 78\, \cr & {\text{Add both equations}} \cr & - 5\,x - 10y = 10\,\,\,\,\, \cr & 5x + 12y = - 78\,\,\,\, \cr & - - - - - - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,\,\,2y\,\, = - 68 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,y = - 34 \cr & {\text{Substitute }}y = 2{\text{ into the equation }}\left( {\bf{1}} \right) \cr & x + \,2\left( { - 34} \right) = - 2 \cr & x - 68 = - 2 \cr & x = 66 \cr & \cr & {\text{The solution set is}} \cr & \left\{ {\left( {66, - 34} \right)} \right\} \cr} $$