To set up the linear equation based on the information provided, we will first express the populations of moose and mountain goats in terms of the number of years, \( n \).
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Population of Moose: The initial population is 200, and it is decreasing by 15 moose each year. So, after \( n \) years, the population of moose can be expressed as: \[ 200 - 15n \]
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Population of Mountain Goats: The initial population is 150, and it is decreasing by 5 mountain goats each year. Therefore, after \( n \) years, the population of mountain goats can be expressed as: \[ 150 - 5n \]
Now, we can set these two expressions equal to each other to find the point in time when the populations will be the same: \[ 200 - 15n = 150 - 5n \]
Next, we will solve this equation for \( n \).
Step 1: Move all terms involving \( n \) to one side
Add \( 15n \) to both sides: \[ 200 = 150 - 5n + 15n \] \[ 200 = 150 + 10n \]
Step 2: Move constant terms to the other side
Subtract 150 from both sides: \[ 200 - 150 = 10n \] \[ 50 = 10n \]
Step 3: Divide both sides by 10
\[ n = \frac{50}{10} = 5 \]
So, it will take 5 years for the populations of moose and mountain goats to be equal.