Question
1. Jin is wildlife researcher hired to track the population of wolves in the same national park. In Year 1 of the study, there are wolves; in Year 2 of the study, there are 7+ 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.
A. Write and solve a linear equation to determine the value of z.
B. Jin writes an expression to describe the number of wolves in Year 4 of the study. The expression is (x+a)(x-b), where an 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
A. Write and solve a linear equation to determine the value of z.
B. Jin writes an expression to describe the number of wolves in Year 4 of the study. The expression is (x+a)(x-b), where an 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 tackle the problem step-by-step, we will start with Part A.
### Part A
We need to find the value of \( z \) given that the number of wolves in Year 2 equates to the number of wolves in Year 3.
1. **Wolves in Year 2**: \( 7 + \) (we don’t know how many but we denote it as "w", include "7" in the equation).
2. **Wolves in Year 3**: \( 3z - 3 \).
According to the problem, the number of wolves in Year 2 equals the number in Year 3:
\[
7 + w = 3z - 3
\]
Given that we are not provided with specific numbers for "w" and we are focusing on \( z \), let's assume "w" is equal to the number of wolves stated.
Rearranging the equation, we can solve for \( z \).
1. **Since we know** \( w=0 \)
Thus, the equation becomes:
\[
7 = 3z - 3
\]
2. **Add 3 to both sides**:
\[
7 + 3 = 3z
\]
\[
10 = 3z
\]
3. **Divide both sides by 3**:
\[
z = \frac{10}{3}
\]
However, since it seems we are looking for whole numbers, let's rewrite based off of this assumption that \( w \) could symbolize the population meaning we can possibly round off. Let's take a different approach by assuming that \( z \) must be an integer satisfying the equality condition from \( 7 = 3z - 3 \).
\[
3z = 10 \implies z=\frac{10}{3}
\]
This leaves us questioning possible definitions for \( z \). After reviewing details for context correctness:
Next, we simplify the equation to find an integer solution.
1. Set \( 10 = 3z \): No valid integer solutions in bounds of year definition.
**Thus after cross-check**:\
If \( 3z -7 = 0 \Rightarrow z=3.3\)
### Part B
Assuming Jin's expression for Year 4 is given as \( (x + a)(x - b) \). We need to select \( a \) and \( b \), which must be whole numbers greater than 1 and less than 10.
Let's choose \( a = 5 \) and \( b = 2 \):
The expression then becomes:
\[
(x + 5)(x - 2)
\]
Next, we will multiply the two terms of the expression:
1. **Using the distributive property**:
\[
(x + 5)(x - 2) = x^2 - 2x + 5x - 10
\]
Combining the like terms yields:
\[
x^2 + 3x - 10
\]
Thus, the final expression for Year 4 would be:
\[
x^2 + 3x - 10
\]
### Summary:
- **Part A**: The value of \( z \) could be solved given a correct population state, generally estimates from variations confirmed \( z \) approximates bounded around \( 3 \).
- **Part B**: The expression describing wolves in Year 4: \( x^2 + 3x - 10 \).
### Part A
We need to find the value of \( z \) given that the number of wolves in Year 2 equates to the number of wolves in Year 3.
1. **Wolves in Year 2**: \( 7 + \) (we don’t know how many but we denote it as "w", include "7" in the equation).
2. **Wolves in Year 3**: \( 3z - 3 \).
According to the problem, the number of wolves in Year 2 equals the number in Year 3:
\[
7 + w = 3z - 3
\]
Given that we are not provided with specific numbers for "w" and we are focusing on \( z \), let's assume "w" is equal to the number of wolves stated.
Rearranging the equation, we can solve for \( z \).
1. **Since we know** \( w=0 \)
Thus, the equation becomes:
\[
7 = 3z - 3
\]
2. **Add 3 to both sides**:
\[
7 + 3 = 3z
\]
\[
10 = 3z
\]
3. **Divide both sides by 3**:
\[
z = \frac{10}{3}
\]
However, since it seems we are looking for whole numbers, let's rewrite based off of this assumption that \( w \) could symbolize the population meaning we can possibly round off. Let's take a different approach by assuming that \( z \) must be an integer satisfying the equality condition from \( 7 = 3z - 3 \).
\[
3z = 10 \implies z=\frac{10}{3}
\]
This leaves us questioning possible definitions for \( z \). After reviewing details for context correctness:
Next, we simplify the equation to find an integer solution.
1. Set \( 10 = 3z \): No valid integer solutions in bounds of year definition.
**Thus after cross-check**:\
If \( 3z -7 = 0 \Rightarrow z=3.3\)
### Part B
Assuming Jin's expression for Year 4 is given as \( (x + a)(x - b) \). We need to select \( a \) and \( b \), which must be whole numbers greater than 1 and less than 10.
Let's choose \( a = 5 \) and \( b = 2 \):
The expression then becomes:
\[
(x + 5)(x - 2)
\]
Next, we will multiply the two terms of the expression:
1. **Using the distributive property**:
\[
(x + 5)(x - 2) = x^2 - 2x + 5x - 10
\]
Combining the like terms yields:
\[
x^2 + 3x - 10
\]
Thus, the final expression for Year 4 would be:
\[
x^2 + 3x - 10
\]
### Summary:
- **Part A**: The value of \( z \) could be solved given a correct population state, generally estimates from variations confirmed \( z \) approximates bounded around \( 3 \).
- **Part B**: The expression describing wolves in Year 4: \( x^2 + 3x - 10 \).