Question

Complete the steps to solve the bike-shop problem.
The owner of a bike shop sells unicycles and bicycles and keeps inventory by counting seats and wheels. One day, she counts 18 seats and 28 wheels. How many of each type of cycle are there?
Define your variables.
Let b = number of
type your answer...
Let u = number of
type your answer...

Answer 1

To solve the bike-shop problem, let's define our variables clearly.

Let:
- \( u \) = number of unicycles
- \( b \) = number of bicycles

Next, we can set up the equations based on the information given:

1. Each unicycle has 1 seat and 1 wheel.
2. Each bicycle has 1 seat and 2 wheels.

From the problem, we know the following:
- The total number of seats is 18.
- The total number of wheels is 28.

Using this information, we can create two equations:

1. For the seats:
\[
u + b = 18
\]

2. For the wheels:
\[
u + 2b = 28
\]

Now we can solve these equations step-by-step.

**Step 1: Solve for one variable using the first equation.**

From the first equation:
\[
u = 18 - b
\]

**Step 2: Substitute this expression into the second equation.**

Substituting \( u \) in the second equation:
\[
(18 - b) + 2b = 28
\]

**Step 3: Simplify and solve for \( b \).**

Simplifying the equation:
\[
18 - b + 2b = 28
\]
\[
18 + b = 28
\]
\[
b = 28 - 18
\]
\[
b = 10
\]

**Step 4: Substitute \( b \) back into the equation for \( u \).**

Now, substitute \( b = 10 \) back into the first equation:
\[
u + 10 = 18
\]
\[
u = 18 - 10
\]
\[
u = 8
\]

**Conclusion:**

There are:
- \( u = 8 \) unicycles
- \( b = 10 \) bicycles

So the final answer is:
- Let \( u = 8 \) (number of unicycles)
- Let \( b = 10 \) (number of bicycles)