Question

Using the properties of determinants, solve the following for x:

$open vertical bar table row cell straight x plus 2 end cell cell space space straight x plus 6 end cell cell space straight x minus 1 end cell row cell straight x plus 6 end cell cell straight x minus 1 end cell cell straight x plus 2 end cell row cell straight x minus 1 end cell cell straight x plus 2 end cell cell straight x plus 6 end cell end table close vertical bar space equals space 0$

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Solution

Let space increment space equals space open vertical bar table row cell straight x plus 2 end cell cell space space straight x plus 6 end cell cell space straight x minus 1 end cell row cell straight x plus 6 end cell cell straight x minus 1 end cell cell straight x plus 2 end cell row cell straight x minus 1 end cell cell straight x plus 2 end cell cell straight x plus 6 end cell end table close vertical bar space equals space 0

Applying space straight C subscript 2 rightwards arrow straight C subscript 2 minus straight C subscript 1 space and space straight C subscript 3 space rightwards arrow space straight C subscript 3 space minus straight C subscript 1 increment space equals space open vertical bar table row cell straight x plus 2 end cell cell space space 4 end cell cell space space minus 3 end cell row cell straight x plus 6 end cell cell space minus 7 end cell cell space minus 4 end cell row cell straight x minus 1 end cell cell space 3 end cell cell space space space 7 end cell end table close vertical bar Applying space straight R subscript 2 space rightwards arrow straight R subscript 2 minus straight R subscript 1 space and space straight R subscript 3 space rightwards arrow straight R subscript 3 minus straight R subscript 1 space space space increment equals space open vertical bar table row cell straight x plus 2 end cell cell space space 4 end cell cell space space minus 3 end cell row 4 cell space minus 11 end cell cell space minus 1 end cell row cell negative 3 end cell cell space minus 1 end cell cell space space 10 end cell end table close vertical bar space Applying space straight R subscript 2 rightwards arrow straight R subscript 2 space plus space straight R subscript 3 increment space equals space open vertical bar table row cell straight x plus 2 end cell cell space space 4 end cell cell space space minus 3 end cell row 1 cell space minus 12 end cell cell space space space 9 end cell row cell negative 3 end cell cell space minus 1 end cell cell space 10 end cell end table close vertical bar Applying space straight R subscript 3 space rightwards arrow straight R subscript 3 plus left parenthesis 3 right parenthesis straight R subscript 2 space increment space equals space open vertical bar table row cell straight x plus 2 end cell cell space 4 end cell cell space minus 3 end cell row 1 cell negative 12 end cell cell space space 9 end cell row 0 cell negative 37 end cell cell space 37 end cell end table close vertical bar Expanding space along space straight C subscript 1

increment space equals space left parenthesis straight x plus 2 right parenthesis space open vertical bar table row cell negative 12 end cell cell space space 9 end cell row cell negative 37 end cell cell space 37 end cell end table close vertical bar minus 1 open vertical bar table row 4 cell space minus 3 end cell row cell negative 37 end cell cell space space 37 end cell end table close vertical bar increment equals left parenthesis straight x plus 2 right parenthesis thin space left parenthesis negative 444 plus 333 right parenthesis minus 1 left parenthesis 148 minus 111 right parenthesis increment equals space left parenthesis straight x plus 2 right parenthesis thin space left parenthesis negative 111 right parenthesis minus 1 left parenthesis 37 right parenthesis therefore space increment space equals space 0 space equals space minus 111 straight x minus 259 therefore space space straight x space equals space minus 259 over 111 equals negative 7 over 3

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