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.(2 points)
200 − ?n = 150 − ?n
It will take ? years for the population of moose and mountain goats to be equal.
The given linear equation is:
200 - 15n = 150 - 5n
To find the number of years it will take for the populations of these two animals to be equal, we can solve the equation.
First, let's simplify the equation:
200 - 150 = -15n + 5n
50 = -10n
Divide both sides by -10:
50 / -10 = -10n / -10
-5 = n
Therefore, it will take 5 years for the population of moose and mountain goats to be equal.
Four times the sum of a number and four is the same as two multiplied by two less than three times the number. What is the solution to this problem?
Let's break down the given sentence into an equation:
"Four times the sum of a number and four" can be represented as: 4(x + 4)
"is the same as" can be represented as: =
"two multiplied by two less than three times the number" can be represented as: 2(3x - 2)
Putting it all together, our equation is:
4(x + 4) = 2(3x - 2)
Now, let's solve for x:
4(x + 4) = 2(3x - 2)
4x + 16 = 6x - 4
Let's isolate the x terms:
4x - 6x = -4 - 16
-2x = -20
Divide both sides by -2:
x = -20 / -2
x = 10
Therefore, the solution to the problem is x = 10.
To complete the linear equation, we need to find the missing numbers in the equation.
Given:
Initial population of moose = 200
Decrease in moose population per year = 15 per year
Initial population of mountain goats = 150
Decrease in mountain goats population per year = 5 per year
The equation for the population of moose is:
200 - 15n
The equation for the population of mountain goats is:
150 - 5n
To find the number of years it will take for the populations of these two animals to be equal, we need to set the equations equal to each other:
200 - 15n = 150 - 5n
Simplifying the equation, we get:
-15n + 5n = 150 - 200
-10n = -50
Dividing both sides of the equation by -10, we find that:
n = 5
Hence, it will take 5 years for the population of moose and mountain goats to be equal.
To find the linear equation that represents the population of moose and mountain goats over time, we need to determine the rates at which each population is decreasing.
Given:
- The population of moose is decreasing by 15 moose each year.
- The population of mountain goats is decreasing by 5 mountain goats each year.
Let's complete the linear equation step by step.
1. Determine the rate of decrease for the moose population:
The moose population is decreasing by 15 each year. So the rate of moose population decrease is -15.
2. Determine the rate of decrease for the mountain goats population:
The mountain goats population is decreasing by 5 each year. So the rate of mountain goats population decrease is -5.
Now we can write the equation:
Population of moose = 200 - 15n
Population of mountain goats = 150 - 5n
To find the number of years it will take for the populations of moose and mountain goats to be equal, we set the two equations equal to each other:
200 - 15n = 150 - 5n
Now we can solve for 'n'.
200 - 15n + 15n = 150 - 5n + 15n
200 = 150 + 10n
200 - 150 = 10n
50 = 10n
n = 50/10
n = 5
So, it will take 5 years for the population of moose and mountain goats to be equal.