A student claims that the gravitational force between two objects depends upon the mass of the objects and the distance between them. Which argument best supports the student's claim?
a- If the distance between two objects increases, the gravitational force between the objects will decrease.
b- If the distance between two objects increases, the gravitational force between the objects will increase.
c- If the distance between two objects decreases, the gravitational force between the objects will disappear.
d- If the distance between two objects decreases, the gravitational force between the objects will not change
a- If the distance between two objects increases, the gravitational force between the objects will decrease.
The sum of two numbers is 22. Three times one number increased by five is 35. What are the two numbers?
Responses
10
12
6
16
11
7
15
The two numbers are 10 and 12.
Explanation:
Let's call the two numbers x and y.
From the first sentence, we know that:
x + y = 22
From the second sentence, we know that:
3x + 5 = 35
Simplifying this equation, we have:
3x = 30
x = 10
Now that we know x is 10, we can substitute it into the first equation:
10 + y = 22
y = 12
So the two numbers are 10 and 12.
At Barnes and Noble, Sylvia purchased a journal and a cookbook that cost a total of $54, not including tax. If the price of the journal, j, is $3 more than 2 times the price of the cookbook, c, which system of linear equations could be used to determine the price of each item?
1. The equation for the total cost
2. The equation for the price of the journal
3. You can choose any method to solve this system: graphing, substitution, or elimination. Choose a method and solve for the price of the journal and the cookbook.
The price of the journal is
The price of the cookbook is
1. The equation for the total cost:
j + c = 54
2. The equation for the price of the journal:
j = 2c + 3
To solve, we can use substitution. Substitute the second equation into the first equation for j:
(2c + 3) + c = 54
Simplify and solve for c:
3c + 3 = 54
3c = 51
c = 17
Now that we know the price of the cookbook is $17, we can use the second equation to find the price of the journal:
j = 2(17) + 3
j = 37
So the price of the journal is $37 and the price of the cookbook is $17.
Samuel currently has 18 rocks in his collection and gains 4 each week. Lewis currently has 30 rocks in his collection and gains 3 each week.
Set up a system of equations to show how many rocks each has in their collection.
Samuel has y=Response areax+Response area.
Lewis has Response areax+Response area.
After how many weeks will Samuel and Lewis have the same number of rocks in their collections? Response area weeks
How many rocks will Samuel and Lewis have when the amount of rocks in their collection is equal? They will each have Response area rocks in their collection.
Samuel has y = 4x + 18 rocks in his collection after x weeks.
Lewis has y = 3x + 30 rocks in his collection after x weeks.
To find when they will have the same number of rocks, we need to set the two equations equal to each other and solve for x:
4x + 18 = 3x + 30
x = 12
So it will take 12 weeks for Samuel and Lewis to have the same number of rocks in their collections.
To find how many rocks they will each have, we can substitute x = 12 into either equation:
Samuel: y = 4(12) + 18 = 66 rocks
Lewis: y = 3(12) + 30 = 66 rocks
So they will each have 66 rocks in their collection when the amount of rocks is equal.
Which graph represents the solution for the equation -5/2x -1 = 4x +2
To solve for x:
-5/2x - 1 = 4x + 2
-5/2x - 4x = 2 + 1
-13/2x = 3
x = -6/13
So the solution is a single point, (-6/13, 0).
The correct graph would be a point, not a line.
Billy is 6 years younger than Amy. The sum of their ages is 22. How old will each of them be in 8 years?
Amy will be Response area and Billy will be Response area.
Let's start by using the given information to write a system of equations:
- Billy is 6 years younger than Amy: B = A - 6
- The sum of their ages is 22: A + B = 22
We can use substitution to solve for their current ages:
- A + (A-6) = 22
- 2A - 6 = 22
- 2A = 28
- A = 14
So Amy is currently 14 years old, and Billy is 14 - 6 = 8 years old.
To find their ages in 8 years, we simply add 8 to each of their current ages:
- Amy: 14 + 8 = 22 years old in 8 years
- Billy: 8 + 8 = 16 years old in 8 years
Therefore, Amy will be 22 years old and Billy will be 16 years old in 8 years.
Solve the system of equations
3x+2y=2
−2x+y=8
Responses
(-4,2)
(-4,2)
(4,-2)
(4,-2)
(-2,4)
(-2,4)
(14,-20)
We can solve this system of equations using either substitution or elimination.
Substitution:
- Solve the second equation for y: y = 2x + 8
- Substitute that expression for y in the first equation: 3x + 2(2x+8) = 2
- Simplify and solve for x: 7x = -14 --> x = -2
- Substitute x = -2 back into either equation to solve for y: y = 2(-2) + 8 = 4
Therefore, the solution is (-2, 4).
Elimination:
- Multiply the second equation by 2: -4x + 2y = 16
- Add that to the first equation: -x + 4y = 18
- Solve for x or y in terms of the other variable: y = (x+18)/4 or x = -4y + 18
- Substitute that expression into either equation to solve for the other variable: 3(-4y+18) + 2y = 2
- Simplify and solve for y: y = 4
- Substitute y = 4 back into either equation to solve for x: -2x + 4 = 8 --> x = -2
Therefore, the solution is (-2, 4).
The answer is (-2,4).
Not including tax, a total of 19 pens and markers cost $11.50. The pens cost $0.25 each, and the markers cost $0.75 each. Write the system of equations that could be used to solve for the number of pens (p) and the number of markers (m) bought.
p+m=Response area
Response areap+ Response aream=Response area
Let's set up two equations based on the given information:
- The total number of pens and markers purchased is 19: p + m = 19
- The total cost of the pens and markers is $11.50: 0.25p + 0.75m = 11.50
Therefore, the system of equations that could be used to solve for the number of pens and markers bought is:
p+m=19
0.25p + 0.75m = 11.50
Joyce wants to mix granola and raisins together to make a snack for her class. Granola costs $2 per pound and raisins cost $4.50 per pound. Joyce is willing to spend $37.50 and wants to make 15 pounds of trail mix.
g+Response area=15
2Response area+4.5r=Response area
Let's set up two equations based on the given information:
- Joyce wants to make 15 pounds of trail mix, so the total weight of granola and raisins combined is 15 pounds: g + r = 15
- Joyce is willing to spend $37.50, so the total cost of the granola and raisins combined is $37.50: 2g + 4.5r = 37.50
Therefore, the system of equations that could be used to solve for the amount of granola and raisins needed is:
g + r = 15
2g + 4.5r = 37.50
Michelle bought some potted plants.
Petunias cost $7 per pot.
Begonias cost $8 per pot.
She bought 22 potted plants and spent a total of $163. How many petunias did she buy?
Responses
7
9
11
14
13
Let's set up a system of equations based on the given information:
- Let p be the number of petunias Michelle bought, and b be the number of begonias Michelle bought.
- We know that Michelle bought a total of 22 potted plants: p + b = 22
- We also know that the total cost of the plants was $163: 7p + 8b = 163
We can solve for either variable using substitution or elimination.
Substitution:
- Solve the first equation for either variable: p = 22 - b
- Substitute that expression for the variable in the second equation: 7(22-b) + 8b = 163
- Simplify and solve for b: b = 9
- Substitute that value back into the first equation to find p: p + 9 = 22 --> p = 13
Therefore, Michelle bought 13 petunias.
Elimination:
- Multiply the first equation by 8: 8p + 8b = 176
- Subtract that equation from the second equation: -p - b = -13
- Solve for either variable: p = 13 - b
- Substitute that expression back into the first equation to solve for the other variable: 13 + b = 22 --> b = 9
Therefore, Michelle bought 13 petunias.
So, the answer is (Option E) 13.
A company owns two dealerships, both of which sell cars and trucks. The first dealership sells a total of 164 cars and trucks. The second dealership sells twice as many cars and half as many trucks as the first dealership, and sells a total of 229 cars and trucks.
An equation for the total cars and trucks for dealership A:
x + y = 164
An equation for the total cars and trucks for dealership B:
2x + 1/2y = 229
How many cars did dealership A sell?
66
How many trucks did the dealership B sell?
Let's set up an equation for the total trucks sold by dealership B. We know that dealership B sold half as many trucks as dealership A, so:
y/2 = total trucks sold by dealership B
We also know that the total number of cars and trucks sold by dealership B is 229, so:
2x + y/2 = 229
We can use substitution to solve for y:
- Solve the first equation for x: x = 164 - y
- Substitute that expression for x in the second equation: 2(164-y) + y/2 = 229
- Simplify and solve for y: y = 90
Therefore, dealership B sold 90/2 = 45 trucks.