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+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.
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
A. To find the value of \( z \), we set the expression for the number of wolves in Year 2 equal to the expression for the number of wolves in Year 3, as given:
\[
7 + z = 3z - 3
\]
Now, let's solve for \( z \):
1. Subtract \( z \) from both sides:
\[
7 = 3z - z - 3
\]
2. Simplify the equation:
\[
7 = 2z - 3
\]
3. Add 3 to both sides:
\[
7 + 3 = 2z
\]
\[
10 = 2z
\]
4. Divide both sides by 2:
\[
z = 5
\]
So, the solution for \( z \) is \( \boxed{5} \).
B. Now, Jin wants to write an expression for the number of wolves in Year 4 and it is given by the expression \( (x+a)(x-b) \), where \( a \) and \( b \) are whole numbers greater than 1 and less than 10.
Let's choose \( a = 3 \) and \( b = 2 \) (both satisfying the given conditions):
The expression for Year 4 would thus be:
\[
(x + 3)(x - 2)
\]
Now, we will multiply the two terms:
1. Apply the distributive property (FOIL method):
\[
(x + 3)(x - 2) = x^2 - 2x + 3x - 6
\]
2. Combine like terms:
\[
x^2 + x - 6
\]
So, the expression that could be describing the number of wolves in Year 4 after multiplying the two terms is:
\[
\boxed{x^2 + x - 6}
\]
\[
7 + z = 3z - 3
\]
Now, let's solve for \( z \):
1. Subtract \( z \) from both sides:
\[
7 = 3z - z - 3
\]
2. Simplify the equation:
\[
7 = 2z - 3
\]
3. Add 3 to both sides:
\[
7 + 3 = 2z
\]
\[
10 = 2z
\]
4. Divide both sides by 2:
\[
z = 5
\]
So, the solution for \( z \) is \( \boxed{5} \).
B. Now, Jin wants to write an expression for the number of wolves in Year 4 and it is given by the expression \( (x+a)(x-b) \), where \( a \) and \( b \) are whole numbers greater than 1 and less than 10.
Let's choose \( a = 3 \) and \( b = 2 \) (both satisfying the given conditions):
The expression for Year 4 would thus be:
\[
(x + 3)(x - 2)
\]
Now, we will multiply the two terms:
1. Apply the distributive property (FOIL method):
\[
(x + 3)(x - 2) = x^2 - 2x + 3x - 6
\]
2. Combine like terms:
\[
x^2 + x - 6
\]
So, the expression that could be describing the number of wolves in Year 4 after multiplying the two terms is:
\[
\boxed{x^2 + x - 6}
\]