Question

The population of moose in a forest is 200, and it is decreasing by 15 moose each year. In the same forest, there are 150 mountain goats. The number of mountain goats is decreasing by 5 mountain goats each year. Let n be the number of years. Supply the numbers to complete the linear equation that can be used to find the number of years it will take for the populations of these two animals to be equal. Then, solve the equation.

Answer 1

To set up the equations for the populations of moose and mountain goats, let's define the populations as functions of time \( n \) (in years).

1. Population of moose:

   • Initial population: 200 moose
   • Decrease per year: 15 moose
   • Therefore, the population of moose after \( n \) years is: \[ P_{\text{moose}} = 200 - 15n \]

2. Population of mountain goats:

   • Initial population: 150 mountain goats
   • Decrease per year: 5 mountain goats
   • Therefore, the population of mountain goats after \( n \) years is: \[ P_{\text{goats}} = 150 - 5n \]

To find out when the populations will be equal, we set the two equations equal to each other: \[ 200 - 15n = 150 - 5n \]

Now, we will solve this equation for \( n \):

1. Rearrange the equation to get all terms involving \( n \) on one side and constant terms on the other side: \[ 200 - 150 = 15n - 5n \] This simplifies to: \[ 50 = 10n \]

2. Now, divide both sides by 10 to isolate \( n \): \[ n = \frac{50}{10} = 5 \]

Thus, it will take 5 years for the populations of moose and mountain goats to be equal.

Verification:

After 5 years:

• Moose population: \[ P_{\text{moose}} = 200 - 15 \times 5 = 200 - 75 = 125 \]
• Mountain goat population: \[ P_{\text{goats}} = 150 - 5 \times 5 = 150 - 25 = 125 \]

Both populations equal 125 after 5 years, confirming our solution is correct.

The final answer is 5 years.

Answer 2

200-___n = 150-____n

Answer 3

To complete the equation \(200 - __n = 150 - __n\) with the correct values related to the populations of moose and mountain goats, we fill in the blanks as follows:

1. The blank for the moose population decrease is 15, since the population of moose is decreasing by 15 each year.
2. The blank for the mountain goat population decrease is 5, since the population of mountain goats is decreasing by 5 each year.

So the completed equation is: \[ 200 - 15n = 150 - 5n \]

In this case, \(15n\) represents the number of moose that have decreased over \(n\) years, and \(5n\) represents the number of mountain goats that have decreased over the same period.