To solve the bike-shop problem, we'll define our variables first:
- Let \( b \) be the number of bicycles.
- Let \( u \) be the number of unicycles.
Then we can formulate the system of equations based on the information provided:
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The total number of seats (bicycles have 2 seats each, while unicycles have 1 seat each): \[ b + u = 18 \]
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The total number of wheels (bicycles have 2 wheels each, while unicycles have 1 wheel each): \[ 2b + u = 28 \]
So the final set of equations for this problem is:
\[ \begin{align*}
- & \quad b + u = 18 \quad \text{(Equation 1)} \
- & \quad 2b + u = 28 \quad \text{(Equation 2)} \end{align*} \]
Now to solve the system of equations:
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From Equation 1, we can express \( u \) in terms of \( b \): \[ u = 18 - b \]
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Substitute \( u = 18 - b \) into Equation 2: \[ 2b + (18 - b) = 28 \]
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Simplifying this: \[ 2b + 18 - b = 28 \ b + 18 = 28 \ b = 10 \]
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Now substitute \( b = 10 \) back into Equation 1 to find \( u \): \[ 10 + u = 18 \ u = 18 - 10 \ u = 8 \]
Thus, the solution to the bike-shop problem is:
- Number of bicycles \( b = 10 \)
- Number of unicycles \( u = 8 \)
Final Answer:
- There are 10 bicycles and 8 unicycles.