Question

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.
Write and solve a linear equation to determine the value of z
.
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 \( z \) based on the information provided:

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

We are given that the number of wolves in Year 2 is equal to the number of wolves in Year 3:
\[
7 + z = 3z - 3
\]

Now, let's solve this equation.

First, we can isolate \( z \):
\[
7 + z + 3 = 3z
\]
\[
10 + z = 3z
\]
Subtract \( z \) from both sides:
\[
10 = 3z - z
\]
\[
10 = 2z
\]
Now, divide both sides by 2:
\[
z = 5
\]

Thus, the value of \( z \) is \( 5 \).

Next, we move on to write an expression for the number of wolves in Year 4. The expression is given as \( (x + a)(x - b) \), where \( a \) and \( b \) are whole numbers greater than 1 and less than 10.

Choosing values for \( a \) and \( b \), we could use \( a = 3 \) and \( b = 4 \) as an example. Thus, the expression becomes:
\[
(x + 3)(x - 4)
\]

Now, let's multiply the two terms of the expression:
\[
(x + 3)(x - 4) = x^2 - 4x + 3x - 12
\]
Combining like terms:
\[
x^2 - x - 12
\]

So, the final expression Jin could use to describe the number of wolves in Year 4 of the study is:
\[
x^2 - x - 12
\]