A thin tube stretched across a street counts the number of pairs of wheels that pass over it. A vehicle classified as type A with two axles registers two counts. A vehicle classified as type B with seven axles registers seven counts. During a 1​-hour ​period, a traffic counter registered 79 counts. How many type A vehicles and type B vehicles passed over the traffic​ counter? List all possible solutions.
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Part 1
Which of the choices below are possible solutions to the​ problem? Select all that apply.
A.
26 type A and 4 type B vehicles
B.
19 type A and 6 type B vehicles
C.
1 type A and 11 type B vehicles
D.
22 type A and 5 type B vehicles
E.
36 type A and 1 type B vehicles
F.
0 type A and 12 type B vehicles
G.
5 type A and 10 type B vehicles
H.
15 type A and 7 type B vehicles
I.
29 type A and 3 type B vehicles
J.
8 type A and 9 type B vehicles

We can solve this problem by setting up a system of equations. Let:

- \( x \) represent the number of type A vehicles (which register 2 counts each),
- \( y \) represent the number of type B vehicles (which register 7 counts each).

The total number of counts is 79, so the equation that represents this is:

\[
2x + 7y = 79
\]

Now, we need to determine all possible integer solutions for \( x \) and \( y \) such that \( x \geq 0 \) and \( y \geq 0 \), while satisfying this equation.

### Solving for \( x \) in terms of \( y \):

\[
2x = 79 - 7y
\]

For \( x \) to be an integer, \( 79 - 7y \) must be even. Since \( 79 \) is odd and \( 7y \) is odd when \( y \) is odd, we can deduce that \( y \) must be odd for \( x \) to be an integer.

Let’s substitute odd values for \( y \) and solve for \( x \).

- For \( y = 1 \):
  \[
  2x = 79 - 7(1) = 72 \quad \Rightarrow \quad x = \frac{72}{2} = 36
  \]

- For \( y = 3 \):
  \[
  2x = 79 - 7(3) = 58 \quad \Rightarrow \quad x = \frac{58}{2} = 29
  \]

- For \( y = 5 \):
  \[
  2x = 79 - 7(5) = 44 \quad \Rightarrow \quad x = \frac{44}{2} = 22
  \]

- For \( y = 7 \):
  \[
  2x = 79 - 7(7) = 30 \quad \Rightarrow \quad x = \frac{30}{2} = 15
  \]

- For \( y = 9 \):
  \[
  2x = 79 - 7(9) = 16 \quad \Rightarrow \quad x = \frac{16}{2} = 8
  \]

- For \( y = 11 \):
  \[
  2x = 79 - 7(11) = 2 \quad \Rightarrow \quad x = \frac{2}{2} = 1
  \]

- For \( y = 12 \):
  \[
  2x = 79 - 7(12) = -5 \quad \text{(not possible since } x \geq 0\text{)}
  \]

Thus, the possible integer solutions are:

\[
(x, y) = (36, 1), (29, 3), (22, 5), (15, 7), (8, 9), (1, 11)
\]

### Matching the possible solutions with the choices:

- A. 26 type A and 4 type B vehicles: \( 2(26) + 7(4) = 52 + 28 = 80 \) (Not possible)
- B. 19 type A and 6 type B vehicles: \( 2(19) + 7(6) = 38 + 42 = 80 \) (Not possible)
- C. 1 type A and 11 type B vehicles: \( 2(1) + 7(11) = 2 + 77 = 79 \) (Possible)
- D. 22 type A and 5 type B vehicles: \( 2(22) + 7(5) = 44 + 35 = 79 \) (Possible)
- E. 36 type A and 1 type B vehicle: \( 2(36) + 7(1) = 72 + 7 = 79 \) (Possible)
- F. 0 type A and 12 type B vehicles: \( 2(0) + 7(12) = 0 + 84 = 84 \) (Not possible)
- G. 5 type A and 10 type B vehicles: \( 2(5) + 7(10) = 10 + 70 = 80 \) (Not possible)
- H. 15 type A and 7 type B vehicles: \( 2(15) + 7(7) = 30 + 49 = 79 \) (Possible)
- I. 29 type A and 3 type B vehicles: \( 2(29) + 7(3) = 58 + 21 = 79 \) (Possible)
- J. 8 type A and 9 type B vehicles: \( 2(8) + 7(9) = 16 + 63 = 79 \) (Possible)

### Final list of correct choices:

- C. 1 type A and 11 type B vehicles
- D. 22 type A and 5 type B vehicles
- E. 36 type A and 1 type B vehicle
- H. 15 type A and 7 type B vehicles
- I. 29 type A and 3 type B vehicles
- J. 8 type A and 9 type B vehicles

Would you like to go over any details, or have questions?

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Here are five relative questions to explore further:

1. How would the solution change if the total counts were different?
2. What if the vehicles had different numbers of axles?
3. How can we use modular arithmetic to solve problems like these?
4. Can we generalize this solution for more vehicle types?
5. How would you solve this using a graph or matrix approach?

**Tip:** For problems involving integer solutions, always check the feasibility of negative values when solving for unknowns.

A thin tube stretched across a street counts the number of pairs of wheels that pass over it. A vehicle classified as type A with two axles registers two counts. A vehicle classified as type B with seven axles registers seven counts. During a 1-hour period, a traffic counter registered 79 counts. How many type A vehicles and type B vehicles passed over the traffic counter? List all possible solutions.

Explore how to solve a system of linear equations where a traffic counter registers 79 counts in 1 hour. This problem involves determining the number of type A and type B vehicles using algebraic methods. Learn about Diophantine equations and how to find integer solutions. Suitable for Grades 9-12 students.

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