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)
Let n be the number of years.
The equation for the moose population is:
200 - 15n
The equation for the mountain goat population is:
150 - 5n
To find the number of years it will take for the populations to be equal, we set the two equations equal to each other and solve for n:
200 - 15n = 150 - 5n
Subtract 150 from both sides:
200 - 150 - 15n = -5n
50 - 15n = -5n
Add 5n to both sides:
50 - 15n + 5n = -5n + 5n
50 - 10n = 0
Subtract 50 from both sides:
50 - 50 - 10n = 0 - 50
-10n = -50
Divide both sides by -10:
-10n / -10 = -50 / -10
n = 5
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 the number be x.
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)
According to the problem, these two expressions are equal. So, we can set up the equation:
4(x + 4) = 2(3x - 2)
Simplifying the equation:
4x + 16 = 6x - 4
Subtracting 4x and adding 4 to both sides:
16 = 2x - 4
Adding 4 to both sides:
20 = 2x
Dividing both sides 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
The Subtraction Property of Equality is performed first in this problem.
Arthur writes the equation a=10b−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)
To rewrite the equation in terms of b, in Arthur's first step, he should add 7 to both sides of the equation.
Tilda writes the equation 5y=20x+8. She wants to rewrite the equation in terms of x. After subtracting 8 from both sides of the equation, what number should she divide both sides of the equation by in order to accomplish this goal?(1 point)
After subtracting 8 from both sides of the equation, Tilda should divide both sides of the equation by 20 in order to rewrite the equation in terms of x.
A string is cut in two so that one section is 5 inches longer than the other. Twice the length of the longer section is three times the length of the shorter section. Write an equation that allows you to solve for the length of the shorter section. How long is the shorter section?(1 point)
Let's assume that the length of the shorter section is x inches.
According to the problem, the longer section is 5 inches longer than the shorter section, so the length of the longer section would be (x + 5) inches.
Twice the length of the longer section is three times the length of the shorter section, which can be represented as:
2(x + 5) = 3x
This equation allows us to solve for the length of the shorter section.
Expanding and simplifying the equation:
2x + 10 = 3x
Subtracting 2x from both sides:
10 = 3x - 2x
Simplifying further:
10 = x
The shorter section is 10 inches long.
Andrea is 6 years younger than Tony. In two years, Tony will be three times as old as Andrea. Let a be Tony’s current age. Which of these linear equations can be used to find Andrea’s and Tony’s current ages?(1 point)
Let's assume Andrea's current age is A and Tony's current age is T.
According to the information given, Andrea is 6 years younger than Tony, so we can say:
A = T - 6
In two years, Tony will be three times as old as Andrea. In two years, Andrea's age will be A+2 and Tony's age will be T+2. So, we can set up the equation:
T + 2 = 3(A + 2)
So, the linear equations that can be used to find Andrea's and Tony's current ages are:
A = T - 6
T + 2 = 3(A + 2)
Five less than three-fourths of x is seven. If x equals 16, what is the last property of equality you would use to justify the solution to this problem?(1 point)
The last property of equality that you would use to justify the solution to this problem would be subtraction property of equality.
Manipulate the equation y=7x−12 so that it is written in terms of x. Which equation is equivalent to y=7x−12?(1 point)
To manipulate the equation y = 7x - 12 so that it is written in terms of x, we can isolate x on one side of the equation.
Adding 12 to both sides:
y + 12 = 7x
Dividing both sides by 7:
(y + 12)/7 = x
Equation equivalent to y = 7x - 12 in terms of x is:
x = (y + 12)/7
Which equation is equivalent to w=x+y−z?(1 point)
The equation equivalent to w = x + y - z is:
w + z = x + y
Look at the expression: −3(2x+7)
(1 point)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
This expression involves these two operations Response area and Response area .
We will use the Response area to simplify this expression.
The simplified expression looks like Response area .
The expression "-3(2x+7)" involves multiplication and addition.
Using the distributive property, the simplified expression is: -6x - 21.
To solve the equation: 24(34+x)=1,200
, give the reasons for each of the given steps.
What was done to the previous step to get to the current equation?
(1 point)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Step 1: 816+24x=1,200
Step 2: 816−816+24x=1,200 − 816
Step 3: 24x=384
Step 4: x=16
To solve the equation 24(34+x) = 1,200, let's go through each step and provide the reasons:
Step 1: 816 + 24x = 1,200
Reason: Use the distributive property to multiply 24 and (34+x) to get the expression on the left side of the equation.
Step 2: 816 - 816 + 24x = 1,200 - 816
Reason: Subtract 816 from both sides of the equation to isolate the term with 24x, and subtract 816 from 1,200.
Step 3: 24x = 384
Reason: Simplify the equation by combining like terms on the left side, resulting in the term 24x.
Step 4: x = 16
Reason: Divide both sides of the equation by 24 to solve for x and isolate it on its own.
As for what was done to the previous step to get to the current equation:
To go from Step 1 to Step 2, we subtracted 816 from both sides of the equation to eliminate the constant term 816 from the left side.
To go from Step 2 to Step 3, we simplified the equation by combining the terms -816 and 816, resulting in 0 on the left side.
To go from Step 3 to Step 4, we divided both sides of the equation by 24 to isolate the variable x.
Simplify: −4(2x+5)
(1 point)
To simplify the expression -4(2x+5), we use the distributive property:
-4 * 2x = -8x
-4 * 5 = -20
Putting it together, the simplified expression is:
-8x - 20
Put the steps in order for solving the equation: −2(4x+8)+2x = −5x+10
(1 point)
The steps to solve the equation -2(4x+8)+2x = -5x+10 in order are as follows:
Step 1: Use the distributive property on -2(4x+8), resulting in -8x - 16 + 2x = -5x + 10.
Step 2: Combine like terms on the left side of the equation, which gives -6x - 16 = -5x + 10.
Step 3: Add 5x to both sides to eliminate the -5x term on the right side, yielding -6x - 16 + 5x = 10.
Step 4: Combine like terms on the left side by simplifying -6x + 5x to -x, resulting in -x - 16 = 10.
Step 5: Add 16 to both sides to isolate -x, giving -x - 16 + 16 = 10 + 16.
Step 6: Simplify -16 + 16 to 0, resulting in -x = 26.
Step 7: Multiply both sides by -1 to solve for x, giving x = -26.
A student solves the following problem: 2(x - 2) + 5x = 24
Step 1: 2x - 4 + 5x = 24
Step 2: 10x - 4 = 24
Step 3: 10x - 4 + 4 = 24 + 4
Step 4: 10x = 28
Step 5: 10x/10=28/10
Step 6: x = 2.8
Where is the mistake? What did the student do incorrectly?
The student made a mistake in Step 3.
In Step 3, they incorrectly added 4 to the right side of the equation without adding it to the left side. It should be:
10x - 4 + 4 = 24 + 4
So the equation becomes:
10x - 4 = 28
After correcting this mistake, the remaining steps are all correct, and the solution is x = 2.8.