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.
200 - n = 150 - n
To solve the equation, we need to isolate the variable.
200 - 150 = -n + n
50 = 0
Since the equation is inconsistent, there is no solution and the populations of moose and mountain goats will never be equal.
200−__n=150−__n
To complete the equation, we need to fill in the missing numbers for the decrease in population each year.
200 - 15n = 150 - 5n
To solve the equation, we need to isolate the variable.
200 - 150 = -5n + 15n
50 = 10n
Divide both sides of the equation by 10.
n = 5
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 problem and translate it into an equation.
Four times the sum of a number and four: 4(x + 4)
Two multiplied by two less than three times the number: 2(3x - 2)
So, the equation becomes: 4(x + 4) = 2(3x - 2)
Let's solve the equation:
4x + 16 = 6x - 4 (distribute on both sides)
16 = 2x - 4 (subtract 4x from both sides)
20 = 2x (add 4 to both sides)
x = 10
So, 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 most efficient method to solve the equation 2(x+3) = 5x - 2 is to first distribute the 2 to the terms inside the parentheses.
So, the step performed first is the Distributive Property (1).
2(x+3)=5x−2
To solve the equation 2(x+3) = 5x - 2, the most efficient method is to first distribute the 2 to the terms inside the parentheses using the Distributive Property.
2(x+3) = 5x - 2
2 * x + 2 * 3 = 5x - 2
2x + 6 = 5x - 2
So, the step performed first is the Distributive Property (1).
2(x+3)=5x−2 solve for x
To solve the equation 2(x+3) = 5x - 2 for x, we'll simplify the equation step by step:
2(x+3) = 5x - 2
2x + 6 = 5x - 2
To isolate the x term, we'll move the constants to the opposite side of the equation:
2x - 5x = -2 - 6
-3x = -8
Next, we'll solve for x by dividing both sides of the equation by -3:
x = -8 / -3
Simplifying the division:
x = 8/3
Therefore, the solution for x is x = 8/3.
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?
To rewrite the equation a = 10b - 7 in terms of b, Arthur should add 7 to both sides of the equation.
a + 7 = 10b - 7 + 7
By adding 7 to both sides, the equation becomes:
a + 7 = 10b
Therefore, Arthur should add 7 to both sides of the equation in his first step.
ilda 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?
After subtracting 8 from both sides of the equation 5y = 20x + 8, Ilda should divide both sides of the equation by 20 in order to rewrite the equation in terms of x.
(5y - 8) / 20 = (20x + 8) / 20
Simplifying:
y - 8/20 = x + 8/20
Further simplifying:
y - 2/5 = x + 2/5
Therefore, Ilda 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)
Responses
20 inches
15 inches
10 inches
12 inches
Let x be the length of the shorter section.
According to the problem, the longer section is 5 inches longer than the shorter section, so the length of the longer section is x + 5.
Twice the length of the longer section is three times the length of the shorter section, can be written as:
2(x + 5) = 3x
Now, we can solve this equation to find the length of the shorter section:
2x + 10 = 3x
Subtract 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
3(a+2)=a−6+2
a+2=3(a+6+2)
a+2=3(a−6+2)
a=3(a+2)
To find Andrea's and Tony's current ages, we can use the equation:
a + 2 = 3(a - 6 + 2)
This equation represents that in two years, Tony's age (a + 2) will be three times as much as Andrea's age (a - 6 + 2).
Therefore, the correct equation is:
a + 2 = 3(a - 6 + 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)
Responses
Division Property of Equality
Identity Property of Multiplication
Subtraction Property of Equality
Addition Property of Equality
The last property of equality that you would use to justify the solution to the problem "Five less than three-fourths of x is seven" is the Addition Property of Equality.
The equation "Five less than three-fourths of x is seven" can be translated to the equation:
(3/4)x - 5 = 7
To solve this equation for x, you would use the Addition Property of Equality to add 5 to both sides of the equation:
(3/4)x - 5 + 5 = 7 + 5
Simplifying:
(3/4)x = 12
Thus, the last property of equality used in this problem is the Addition 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=(y7)+12
x=(y12)−7
x=(y+12)7
x=(y−12)7
To manipulate the equation y = 7x - 12 so that it is written in terms of x, we need to solve for x.
Adding 12 to both sides of the equation gives us:
y + 12 = 7x
Next, we divide both sides of the equation by 7:
(y + 12)/7 = x
Therefore, the equivalent equation to y = 7x - 12 in terms of x is:
x = (y + 12)/7
Which equation is equivalent to w=x+y−z?(1 point)
Responses
z=w−x−y
z=x+y−w
z=x+y+w
z=(x+y)w
To rewrite the equation w = x + y − z in terms of z, we need to isolate the variable z.
To do this, we rearrange the equation by subtracting x and y from both sides:
w − x − y = -z
To isolate z, we multiply both sides of the equation by -1:
-(w − x − y) = z
Simplifying:
z = -w + x + y
Therefore, the equation equivalent to w = x + y − z, in terms of z, is:
z = -w + x + y.
So, the correct response is z = -w + x + y.
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 .
I apologize, but there seems to be an error in your question. It seems that you have not provided the expression or the operations involved. Please provide the expression and the operations involved, and I will be happy to assist you in simplifying it.
Look at the expression: −3(2x+7)
This expression involves these two operations ____and ____.
We will use the _____to simplify this expression.
The simplified expression looks like _____ .
The expression −3(2x+7) involves the operations of multiplication and addition.
We will use the distributive property to simplify this expression.
The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac.
Applying the distributive property to the expression −3(2x+7), we get:
−3(2x+7) = −3 * 2x + −3 * 7
Simplifying further:
−6x + −21
So, 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?
To solve the equation: 24(34+x) = 1,200, we will break down the steps and provide the reasoning for each step:
Step 1: Apply the distributive property by multiplying 24 by each term inside the parentheses.
Reason: The distributive property states that a(b + c) = ab + ac, so we distribute 24 to both 34 and x.
Step 2: Simplify the expression on the left side of the equation.
Reason: We multiply 24 by 34 and 24 by x to perform the multiplication and obtain a simplified expression.
Step 3: Rewrite the equation with the simplified expression on the left side.
Reason: We want to simplify the equation and have the terms on the left side and the constant value of 1,200 on the right side.
Step 4: Combine like terms on the left side of the equation.
Reason: We simplify the left side further by adding the products of 24 and 34 and 24 and x.
The previous step involved applying the distributive property to simplify the expression inside the parentheses.
Simplify: −4(2x+5)
To simplify the expression −4(2x+5), we will apply the distributive property by multiplying −4 with each term inside the parentheses.
−4(2x+5) = −4 * 2x + −4 * 5
Simplifying further:
−8x + −20
So, the simplified expression is −8x − 20.
Put the steps in order for solving the equation: −2(4x+8)+2x = −5x+10
To solve the equation: −2(4x+8)+2x = −5x+10, we need to simplify the equation step by step. Here are the steps in order:
Step 1: Apply the distributive property to the term −2(4x+8):
-2 * 4x + -2 * 8 + 2x = −5x + 10
Step 2: Simplify the expression on both sides:
-8x - 16 + 2x = -5x + 10
Step 3: Combine like terms on both sides:
-6x - 16 = -5x + 10
Step 4: Add 5x to both sides to isolate the x term:
-6x + 5x - 16 = -5x + 5x + 10
Step 5: Simplify the expression on both sides:
-x - 16 = 10
Step 6: Add 16 to both sides to isolate the x term:
-x - 16 + 16 = 10 + 16
Step 7: Simplify the expression on both sides:
-x = 26
Step 8: Multiply both sides by -1 to solve for x:
(-1) * -x = 26 * -1
Step 9: Simplify the expression on both sides:
x = -26
So, the correct order of steps for solving the equation is:
1. Apply the distributive property.
2. Simplify both sides.
3. Combine like terms on both sides.
4. Add or subtract to isolate the x term.
5. Simplify both sides.
6. Add or subtract to isolate the x term.
7. Simplify both sides.
8. Multiply both sides.
9. Simplify both sides.
Therefore, the steps in order for solving the equation are: 1, 2, 3, 4, 5, 6, 7, 8, 9.
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 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 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 of the equation instead of subtracting it.
Step 3: 10x - 4 + 4 = 24 + 4
The correct step would be:
Step 3: 10x - 4 - 4 = 24 - 4
By subtracting 4 from both sides, the equation becomes:
10x - 4 = 20
The subsequent steps would then be:
Step 4: 10x = 20 + 4
Step 5: 10x = 24
Step 6: x = 24/10
Simplifying further:
x = 2.4
Therefore, the correct response is: Step 3: The student should have subtracted 4 from both sides, not added 4.