. Roger is a wildlife researcher who has been hired to study and track the
population of bears in a national park. In Year 1 of the study, Roger determines
that the park is home to x bears. In Year 2, the population has increased to
x+10 bears, and in Year 3, the population has increased to 2(x+10) bears.
a. Use the Distributive Property to rewrite the bear population in Year 3
without needing to use parentheses
. To rewrite the bear population in Year 3 without parentheses, we can distribute the 2 to both x and 10:
2(x+10) = 2x + 20
Therefore, the bear population in Year 3 can be rewritten as 2x + 20 bears.
The total number of bears in the park during Year 2 is 42. Write and solve a
linear equation to determine how many bears were in the park during Year
1. Explain the answer
Given that the bear population in Year 2 is x+10 bears and that this number is equal to 42, we can write the following equation:
x + 10 = 42
Subtracting 10 from both sides to isolate x, we get:
x = 42 - 10
x = 32
Therefore, there were 32 bears in the park during Year 1. This can be verified by looking back at the information provided in the question, where it was mentioned that in Year 1, the population of bears was x bears. The given value of x (32) confirms this answer.
The total number of bears in the park during Year 3 is 50. Write and solve a
linear equation to determine how many bears were in the park during Year
1. Note that this answer will not be the same as the answer you got in part
b.
In Year 3, the bear population is given by 2(x+10), and we know that the total number of bears in the park during Year 3 is 50. Therefore, we can write the following equation:
2(x+10) = 50
Expanding the left side of the equation, we get:
2x + 20 = 50
Subtracting 20 from both sides to isolate x, we get:
2x = 30
x = 15
Therefore, there were 15 bears in the park during Year 1. This is different from the result we obtained in the previous part because the increase in the bear population from Year 1 to Year 2 was different in this scenario.
. During Year 4, bad weather keeps Roger from finding the exact number of
bears in the park. However, he knows that y+5>38, where y is the number
of bears in the park during Year 4. Solve the inequality, and give three
possible values for y.
To solve the inequality y + 5 > 38, we first need to isolate y on one side of the inequality sign.
Subtracting 5 from both sides gives us:
y > 38 - 5
y > 33
Therefore, y must be greater than 33.
Now, three possible values for y that satisfy this inequality are:
- y = 34
- y = 35
- y = 36
These are all solutions because each value satisfies the condition that y is greater than 33.
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.
a. Write and solve a linear equation to determine the value of z
Given:
- In Year 1, there are z wolves.
- In Year 2, there are 7+z wolves.
- In Year 3, there are 3z-3 wolves.
Since the number of wolves in Year 2 is equal to the number of wolves in Year 3, we can set up the following equation:
7 + z = 3z - 3
Solving for z:
7 + z = 3z - 3
Add 3 to both sides:
10 + z = 3z
Subtract z from both sides:
10 = 2z
Divide by 2:
z = 5
Therefore, the value of z is 5.