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A shop sells bicycles and tricycles. In total, there are 7 cycles (this includes both bicycles and tricycles) and 19 wheels. Determine how many of each there are, if a bicycle has two wheels and a tricycle has three wheels.
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Let $x$ and $y$ be numbers of bicycles and tricycles, respectively

$x + y = 7$               (i)

$2x + 3y = 19$        (ii)

Multiplying (1) by 2 gives: $2x + 2y = 14$        (iii)

Subtracting (iii) from (ii) gives:

$y = 19 - 14 = 5$

Substituting 5 for $y$ in (i) gives: $x = 7 - 5 = 2$

Therefore there are 2 bicycles and 5 tricycles.
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