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Determinants

Question

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Determine x so that the points (3, 2), (x, 2) and (8, 8) lie on a line.

Solution

Since the points (3, – 2), (x, 2) and (8, 8) lie on a line area of the triangle formed by the points is zero.

therefore space space space 1 half open vertical bar table row 3 cell space space minus 2 end cell cell space space 1 end cell row straight x cell space 2 end cell cell space 1 end cell row 8 cell space 8 end cell cell space space 1 end cell end table close vertical bar space equals space 0 space space space space space rightwards double arrow space space space space space space space open vertical bar table row 3 cell negative 2 end cell 1 row straight x 2 1 row 8 8 1 end table close vertical bar space equals space 0 rightwards double arrow space space open vertical bar table row 3 cell space space minus 2 end cell cell space space 1 end cell row cell straight x minus 3 end cell cell space space 4 end cell cell space space space 0 end cell row 5 cell space 10 end cell cell space space space 0 end cell end table close vertical bar space equals space 0 comma space space space by space straight R subscript 2 space minus straight R subscript 1 comma space space straight R subscript 3 space minus straight R subscript 1 rightwards double arrow space open vertical bar table row cell straight x minus 3 end cell cell space space space space space 4 end cell row 5 cell space space space 10 end cell end table close vertical bar space equals space 0 comma space expanding space with space third space column space straight C subscript 3. rightwards double arrow space space 10 left parenthesis straight x minus 3 right parenthesis minus 20 space equals space 0 space space space space rightwards double arrow space space straight x minus 3 minus 2 space equals 0 space space space space rightwards double arrow space space space straight x space equals space 5.

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