STEP 1:
\(6x+5xy-5y=8\\ \\ 6x+5y\left( x-1 \right) =8\\ \\ 5y\left( x-1 \right) =8-6x\\ \\ 5y\left( x-1 \right) =2\left( 4-3x \right) \\ \\ \frac { 1 }{ 5\left( x-1 \right) } \cdot 5y\left( x-1 \right) =2\left( 4-3x \right) \cdot \frac { 1 }{ 5\left( x-1 \right) } \\ \\ y=\frac { 2\left( 4-3x \right) }{ 5\left( x-1 \right) }\)
STEP 2:
\(x+y=3\\ \\ y=3-x\)
STEP 3:
This means that...
\(3-x=\frac { 2\left( 4-3x \right) }{ 5\left( x-1 \right) } \\ \\ \left\{ 5\left( x-1 \right) \right\} \left( 3-x \right) =2\left( 4-3x \right) \\ \\ \left( 5x-5 \right) \left( 3-x \right) =8-6x\\ \\ 15x-5{ x }^{ 2 }-15+5x=8-6x\\ \\ 20x-5{ x }^{ 2 }-15=8-6x\\ \\ 5{ x }^{ 2 }+8-6x+15-20x=0\\ \\ 5{ x }^{ 2 }-26x+23=0\)
STEP 4:
Which means that...
\(\\ \frac { 1 }{ 5 } \cdot \left( 5{ x }^{ 2 }-26x+23 \right) =0\cdot \frac { 1 }{ 5 } \\ \\ { x }^{ 2 }-\frac { 26 }{ 5 } x+\frac { 23 }{ 5 } =0\\ \\ { x }^{ 2 }-\frac { 26 }{ 5 } x=-\frac { 23 }{ 5 }\)
Which means that...
\(\ \\ { \left( x-\frac { 13 }{ 5 } \right) }^{ 2 }-{ \left( \frac { 13 }{ 5 } \right) }^{ 2 }=-\frac { 23 }{ 5 } \\ \\ { \left( x-\frac { 13 }{ 5 } \right) }^{ 2 }-\frac { 169 }{ 25 } =-\frac { 23 }{ 5 } \\ \\ { \left( x-\frac { 13 }{ 5 } \right) }^{ 2 }=-\frac { 115 }{ 25 } +\frac { 169 }{ 25 } \\ \\ { \left( x-\frac { 13 }{ 5 } \right) }^{ 2 }=\frac { 54 }{ 25 }\)
Which means that...
\(\\ \\ x-\frac { 13 }{ 5 } =\pm \frac { \sqrt { 54 } }{ 5 } =\pm \frac { 3\sqrt { 6 } }{ 5 } \\ \\ \therefore \quad x=\frac { 13 }{ 5 } \pm \frac { 3\sqrt { 6 } }{ 5 }\)
STEP 5:
When:
\(x=\frac { 13 }{ 5 } +\frac { 3\sqrt { 6 } }{ 5 }\)
.........
\(y=3-\left( \frac { 13 }{ 5 } +\frac { 3\sqrt { 6 } }{ 5 } \right) \\ \\ y=\frac { 15 }{ 5 } -\frac { 13 }{ 5 } -\frac { 3\sqrt { 6 } }{ 5 } \\ \\ y=\frac { 15-13-3\sqrt { 6 } }{ 5 } \\ \\ y=\frac { 2-3\sqrt { 6 } }{ 5 }\)
STEP 6:
When :
\(x=\frac { 13 }{ 5 } -\frac { 3\sqrt { 6 } }{ 5 }\)
...........
\(y=3-\left( \frac { 13 }{ 5 } -\frac { 3\sqrt { 6 } }{ 5 } \right) \\ \\ y=\frac { 15 }{ 5 } -\frac { 13 }{ 5 } +\frac { 3\sqrt { 6 } }{ 5 } \\ \\ y=\frac { 15-13+3\sqrt { 6 } }{ 5 } \\ \\ y=\frac { 2+3\sqrt { 6 } }{ 5 }\)
