Question

2. Jin is a wildlife researcher hired to track the population of wolves in the same national park. In Year 1 of the study, there are z wolves; in Year 2 of the study, there are 7+z wolves; and in Year 3 of the study, there are 3z-3 wolves. The number of wolves in Year 2 is equal to the number of wolves in Year 3.

Jin writes an expression to describe the number of wolves in Year 4 of the study. The expression is (x+a)(x-b), where a and b are whole numbers greater than 1 and less than 10. Write an expression that could be the expression Jin uses to describe the number of wolves in the park during Year 4 of the study. Then multiply the two terms of the expression.

Answer 1

To solve for the number of wolves in Year 1, Year 2, and Year 3, we start with the equations given:

- Year 1: \( z \)
- Year 2: \( 7 + z \)
- Year 3: \( 3z - 3 \)

According to the problem, the number of wolves in Year 2 is equal to the number of wolves in Year 3:

\[
7 + z = 3z - 3
\]

Now, we can solve for \( z \):

1. Rearranging gives:
\[
7 + z + 3 = 3z
\]
\[
10 = 3z - z
\]
\[
10 = 2z
\]
2. Dividing both sides by 2:
\[
z = 5
\]

Now we can find the number of wolves each year:

- Year 1: \( z = 5 \)
- Year 2: \( 7 + z = 7 + 5 = 12 \)
- Year 3: \( 3z - 3 = 3(5) - 3 = 15 - 3 = 12 \)

Both Year 2 and Year 3 have 12 wolves, confirming our calculations.

Next, we need to write an expression for Year 4 of the study, given as \( (x + a)(x - b) \), where \( a \) and \( b \) are whole numbers greater than 1 and less than 10. Let's choose:

- \( a = 3 \)
- \( b = 2 \)

Thus, the expression for Year 4 would be:

\[
(x + 3)(x - 2)
\]

Now, we will multiply the two terms:

\[
(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6
\]

Therefore, the expression Jin uses to describe the number of wolves in Year 4 of the study is:

\[
x^2 + x - 6
\]