Question

Solve for x and y in the following set of equation

6x+5xy-5y=8
X+y=3

Please help this is important

Answer 1

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 }`