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. 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.
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.
Answers
GPT-4o mini
To solve for the number of wolves, let's start by writing down the given conditions.
1. In Year 1, there are \( z \) wolves.
2. In Year 2, there are \( 7 + z \) wolves.
3. In Year 3, there are \( 3z - 3 \) wolves.
According to the problem, the number of wolves in Year 2 is equal to the number of wolves in Year 3:
\[
7 + z = 3z - 3
\]
To solve for \( z \), we can rearrange the equation:
\[
7 + z + 3 = 3z
\]
This simplifies to:
\[
10 = 3z - z
\]
\[
10 = 2z
\]
Dividing both sides by 2:
\[
z = 5
\]
Now that we know \( z \), we can find the number of wolves each year:
- Year 1: \( z = 5 \)
- Year 2: \( 7 + z = 7 + 5 = 12 \)
- Year 3: \( 3z - 3 = 3(5) - 3 = 15 - 3 = 12 \)
Now we have confirmed that the number of wolves in Years 2 and 3 is indeed equal (12 wolves each).
Next, let's create an expression to describe the number of wolves in Year 4.
The problem states that the Year 4 expression is of the form \( (x + a)(x - b) \) where \( a \) and \( b \) are whole numbers greater than 1 and less than 10. Let’s select values for \( a \) and \( b \).
Let's choose \( a = 3 \) and \( b = 2 \). Then our expression becomes:
\[
(x + 3)(x - 2)
\]
Now we will multiply these two terms:
\[
(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6
\]
Thus, the expression that Jin uses to describe the number of wolves in Year 4 of the study is:
\[
x^2 + x - 6
\]
1. In Year 1, there are \( z \) wolves.
2. In Year 2, there are \( 7 + z \) wolves.
3. In Year 3, there are \( 3z - 3 \) wolves.
According to the problem, the number of wolves in Year 2 is equal to the number of wolves in Year 3:
\[
7 + z = 3z - 3
\]
To solve for \( z \), we can rearrange the equation:
\[
7 + z + 3 = 3z
\]
This simplifies to:
\[
10 = 3z - z
\]
\[
10 = 2z
\]
Dividing both sides by 2:
\[
z = 5
\]
Now that we know \( z \), we can find the number of wolves each year:
- Year 1: \( z = 5 \)
- Year 2: \( 7 + z = 7 + 5 = 12 \)
- Year 3: \( 3z - 3 = 3(5) - 3 = 15 - 3 = 12 \)
Now we have confirmed that the number of wolves in Years 2 and 3 is indeed equal (12 wolves each).
Next, let's create an expression to describe the number of wolves in Year 4.
The problem states that the Year 4 expression is of the form \( (x + a)(x - b) \) where \( a \) and \( b \) are whole numbers greater than 1 and less than 10. Let’s select values for \( a \) and \( b \).
Let's choose \( a = 3 \) and \( b = 2 \). Then our expression becomes:
\[
(x + 3)(x - 2)
\]
Now we will multiply these two terms:
\[
(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6
\]
Thus, the expression that Jin uses to describe the number of wolves in Year 4 of the study is:
\[
x^2 + x - 6
\]