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.

Question

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:

\[ x^2 + x - 6 \]