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 equation that can be used to find the number of years it will take for the populations of the two animals to be equal is:
200 - 15n = 150 - 5n
To solve the equation, we can begin by simplifying it:
200 - 15n = 150 - 5n
-15n + 5n = 150 - 200
-10n = -50
n = (-50)/(-10)
n = 5
Therefore, it will take 5 years for the population of moose and mountain goats to be equal.
To complete the equation and solve it, we need to plug in the values for the decrease in population each year for both the moose and the mountain goats.
The equation is:
200 - 15n = 150 - 5n
Now, let's solve it:
-15n + 5n = 150 - 200
-10n = -50
n = 5
Therefore, it will take 5 years for the population of moose and mountain goats to be equal.
Let's break down the problem into an equation.
Four times the sum of a number and four can be expressed as: 4(x + 4)
Two multiplied by two less than three times the number can be expressed as: 2(3x - 2)
According to the given information, these two expressions are equal:
4(x + 4) = 2(3x - 2)
Now let's solve for x:
4x + 16 = 6x - 4
4x - 6x = -4 - 16
-2x = -20
Dividing both sides by -2:
x = -20 / -2
x = 10
Therefore, the solution to the problem is x = 10.
When using the most efficient method to solve the equation 2(x + 3) = 5x - 2, the first step to be performed is 1 for Distributive Property.
In order to rewrite the equation a = 10b - 7 in terms of b, Arthur should first add 7 to both sides of the equation.
After subtracting 8 from both sides of the equation 5y = 20x + 8, Tilda should divide both sides of the equation by 20 in order to rewrite the equation in terms of x.
Let's denote the length of the shorter section as x inches.
According to the given information, the longer section is 5 inches longer than the shorter section, which means its length is x + 5 inches.
Twice the length of the longer section is three times the length of the shorter section can be expressed as:
2(x + 5) = 3x
Now, we can solve this equation to find the length of the shorter section:
2x + 10 = 3x
Subtracting 2x from both sides:
10 = x
Therefore, the length of the shorter section is 10 inches.
200−
n=150−
n
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)
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
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?
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?
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)
Responses
20 inches
10 inches
15 inches
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)
Responses
20 inches
10 inches
15 inches
12 inches
Let's denote the length of the shorter section as x inches.
According to the given information, the longer section is 5 inches longer than the shorter section, which means its length is x + 5 inches.
Twice the length of the longer section is three times the length of the shorter section can be expressed as:
2(x + 5) = 3x
Now, we can solve this equation to find the length of the shorter section:
2x + 10 = 3x
Subtracting 2x from both sides:
10 = x
Therefore, the length of the shorter section is 10 inches.
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)
Responses
a=3(a+2)
a equals 3 left parenthesis a plus 2 right parenthesis
a+2=3(a+6+2)
a plus 2 equals 3 left parenthesis a plus 6 plus 2 right parenthesis
3(a+2)=a−6+2
3 left parenthesis a plus 2 right parenthesis equals a minus 6 plus 2
a+2=3(a−6+2)
The correct linear equation that can be used to find Andrea's and Tony's current ages is:
a + 2 = 3(a - 6 + 2).
This equation takes into account that Andrea is 6 years younger than Tony and that in two years, Tony will be three times as old as Andrea.
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)
Responses
Addition Property of Equality
Addition Property of Equality
Subtraction Property of Equality
Subtraction Property of Equality
Identity Property of Multiplication
Identity Property of Multiplication
Division Property of Equality
The last property of equality you would use to justify the solution to the problem "Five less than three-fourths of x is seven" when x equals 16 is the 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)
Responses
x=(y+12)7
x equals Start Fraction left parenthesis y plus 12 right parenthesis over 7 End Fraction
x=(y12)−7
x equals left parenthesis Start Fraction y over 12 End Fraction right parenthesis minus 7
x=(y−12)7
x equals Start Fraction left parenthesis y minus 12 right parenthesis over 7 End Fraction
x=(y7)+12
To write the equation y = 7x - 12 in terms of x, we need to isolate x on one side of the equation.
x = (y + 12)/7
Therefore, the equation that is equivalent to y = 7x - 12 is x = (y + 12)/7.
Which equation is equivalent to w=x+y−z?(1 point)
Responses
z=x+y−w
z equals x plus y minus w
z=w−x−y
z equals w minus x minus y
z=(x+y)w
z equals Start Fraction left parenthesis x plus y right parenthesis over w End Fraction
z=x+y+w
The equation that is equivalent to w = x + y - z is z = w - 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 .
This expression involves the operations of multiplication and addition.
We will use the distributive property to simplify this expression.
The simplified expression looks like -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) = 1200, let's go through each step and explain the reasons for the given steps:
Step 1: 816 + 24x = 1200
In this step, we distribute the 24 to both terms inside the parentheses using the distributive property.
Step 2: 816 - 816 + 24x = 1200 - 816
Here, we combine like terms by subtracting 816 from both sides of the equation.
Step 3: 24x = 384
In this step, we simplify the expression on the right side of the equation by subtracting 816 from 1200, which gives us 384.
Step 4: x = 16
Finally, we divide both sides of the equation by 24 to solve for the variable x.
Therefore, the reason for each step is as follows:
Step 1: Distributive property
Step 2: Combine like terms
Step 3: Simplify the expression on one side of the equation
Step 4: Solve for x by dividing both sides by 24
The previous step involved subtracting 816 from both sides of the equation to simplify the expression, resulting in the current equation 816 - 816 + 24x = 1200 - 816.
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 get to the current equation from the previous step, the following was done:
Step 1: 816 + 24x = 1200 (Initial equation)
Step 2: 816 - 816 + 24x = 1200 - 816
In this step, 816 was subtracted from both sides of the equation to simplify the expression.
Step 3: 24x = 384
This step involved simplifying the expression on the right side of the equation by subtracting 816 from 1200, resulting in 384.
Step 4: x = 16
Finally, the equation was solved for x by dividing both sides by 24.
Therefore, the step that was done to the previous step to get to the current equation is subtracting 816 from both sides of the equation.
Simplify: −4(2x+5)
(1 point)
Responses
(−4÷2x)+(−4÷5)
open paren negative 4 divided by 2 x close paren plus open paren negative 4 divided by 5 close paren
(−4+2x)×(−4+5)
open paren negative 4 plus 2 x close paren times open paren negative 4 plus 5 close paren
(−4×2x)+(−4×5)
open paren negative 4 times 2 x close paren plus open paren negative 4 times 5 close paren
(−4−2x)+(−4−5)
To simplify the expression -4(2x+5), we need to distribute the -4 to both terms inside the parentheses.
The simplified expression is: (-4)(2x) + (-4)(5) = -8x - 20.
Therefore, the correct response is:
(-4×2x)+(−4×5)
Put the steps in order for solving the equation: −2(4x+8)+2x = −5x+10
(1 point)
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The correct order for solving the equation -2(4x+8) + 2x = -5x + 10 is as follows:
Step 1: Distribute -2 to both terms inside the parentheses: -8x - 16 + 2x = -5x + 10.
Step 2: Combine like terms on each side of the equation: -6x - 16 = -5x + 10.
Step 3: Move all terms with x to one side by adding 6x to both sides: -16 = x + 10.
Step 4: Move the constant term to the other side by subtracting 10 from both sides: -26 = x.
Therefore, the correct order for solving the equation is:
1. Distribute -2 to both terms inside the parentheses.
2. Combine like terms on each side of the equation.
3. Move all terms with x to one side.
4. Move the constant term to the other side.
Note: It seems that there is a missing 2x term in your provided equation. Please double-check the equation for accuracy.
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?
(1 point)
Responses
Step 5: The student should have multiplied by 10 on both sides, not divided by 10.
Step 5: The student should have multiplied by 10 on both sides, not divided by 10.
Step 2: The student should have added 2x + 5x = 7x, not (2x)(5) = 10x.
Step 2: The student should have added 2x + 5x = 7x, not (2x)(5) = 10x.
Step 3: The student should have subtracted 4 from both sides, not added 4.
Step 3: The student should have subtracted 4 from both sides, not added 4.
Step 1: The student should have only distributed the 2 and x, not the x & -2.
The mistake made by the student is in Step 3: The student incorrectly added 4 to both sides instead of subtracting 4.
In Step 1, the student correctly distributed the 2 to both x and -2.
In Step 2, the student correctly combined like terms and simplified the expression.
However, in Step 3, instead of subtracting 4 from both sides, the student mistakenly added 4 to both sides.
To correct the equation, the student should have subtracted 4 from both sides:
Step 3: 10x - 4 - 4 = 24 - 4
10x - 8 = 20
Following this correction, the rest of the steps would be:
Step 4: 10x = 20 + 8
10x = 28
Step 5: (10x)/10 = 28/10
x = 2.8
Therefore, the correct response is:
Step 3: The student should have subtracted 4 from both sides, not added 4.