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The population of moose in the 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
be equal.
years for the population of moose and mountain goats to

7 answers

To find when the populations of moose and mountain goats will be equal, we can set up equations for each population over time.

  1. Moose Population: The starting population is 200 and it decreases by 15 each year. So, the equation for the moose population after \( n \) years is: \[ 200 - 15n \]

  2. Mountain Goat Population: The starting population is 150 and it decreases by 5 each year. So, the equation for the mountain goat population after \( n \) years is: \[ 150 - 5n \]

Next, we want to find \( n \) when these populations are equal: \[ 200 - 15n = 150 - 5n \]

Now, we can solve this equation.

  1. Rearrange the equation: \[ 200 - 150 = 15n - 5n \] \[ 50 = 10n \]

  2. Divide both sides by 10: \[ n = 5 \]

So, it will take 5 years for the populations 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?
(1 point)

Let's define the unknown number as \( x \).

According to the problem:

Four times the sum of a number and four can be represented as: \[ 4(x + 4) \]

Two multiplied by two less than three times the number can be represented as: \[ 2(3x - 2) \]

Now we can set up the equation: \[ 4(x + 4) = 2(3x - 2) \]

Next, we can simplify both sides of the equation:

  1. Expand the left side: \[ 4x + 16 \]

  2. Expand the right side: \[ 6x - 4 \]

Now, we rewrite the equation: \[ 4x + 16 = 6x - 4 \]

Now, let's isolate \( x \):

  1. Subtract \( 4x \) from both sides: \[ 16 = 6x - 4x - 4 \] \[ 16 = 2x - 4 \]

  2. Add \( 4 \) to both sides: \[ 20 = 2x \]

  3. Divide by \( 2 \): \[ x = 10 \]

The solution to the problem is \( x = 10 \).

When using the most efficient method to solve the following problem, what step is performed first? Enter 1 for Distributive Property or 2 for Subtraction
Property of Equality.
2(x+3) = 5x-2
(1 point)

To solve the equation \(2(x + 3) = 5x - 2\) using the most efficient method, we should first apply the Distributive Property to the left-hand side of the equation.

Thus, the correct answer is:

1 for Distributive Property.

Arthur writes the equation a = 106 - 7. He wants to rewrite it in terms of b.
In Arthur's first step, what value should he add to both sides of the equation? (1 point)

Arthur's equation is

\[ a = 106 - 7. \]

First, let's simplify the right side:

\[ a = 99. \]

If Arthur wants to rewrite this equation in terms of \( b \), it is common to express \( b \) in terms of \( a \).

Suppose he wants to form a new equation involving \( b \) such that it is dependent on \( a \). One common method could be to express \( b \) as a function of \( a \). To do this, let's assume he wants to relate \( b = a \).

If Arthur wishes to add a value to both sides in order to maintain equality while rewriting in terms of \( b \), he could decide on a value of \( b \). For example, if he wanted \( b \) to equal \( a + k \) for some constant \( k \), he would obviously need to add \( k \) to both sides.

However, without additional context on how \( b \) is intended to relate to \( a \), specifically determining what value should be added is ambiguous.

If we want the equation to remain consistent and maintain equality while allowing for a new variable, the safe approach could be to manipulate \( a \) to \( b \) without change, meaning Arthur might consider adding \( 0 \).

But let's clarify: if he was trying to move to a new value, he could consider adding \( 7 \) to adjust for \( b \) being isolated elsewhere.

Given this context, please clarify what relationship \( b \) has to \( a \), or if there are constraints on the form it takes. Otherwise, if no changes are intended, the added value could simply be \( 0\).