Question
One method for solving 3(xβ4)=18
3
(
π₯
β
4
)
=
18
is to first divide by 3, then add 4. Which method below would also result in the correct answer?(1 point)
Responses
First, distribute the 3, then add 12, and lastly divide by 3.
First, distribute the 3, then add 12, and lastly divide by 3.
First, distribute the 3, then add 4, and lastly divide by 3.
First, distribute the 3, then add 4, and lastly divide by 3.
First, divide by 3, then subtract 4.
First, divide by 3, then subtract 4.
First, multiply by 1/3
1
3
, then subtract 4.
First, multiply by Start Fraction 1 over 3 End Fraction , then subtract 4.
3
(
π₯
β
4
)
=
18
is to first divide by 3, then add 4. Which method below would also result in the correct answer?(1 point)
Responses
First, distribute the 3, then add 12, and lastly divide by 3.
First, distribute the 3, then add 12, and lastly divide by 3.
First, distribute the 3, then add 4, and lastly divide by 3.
First, distribute the 3, then add 4, and lastly divide by 3.
First, divide by 3, then subtract 4.
First, divide by 3, then subtract 4.
First, multiply by 1/3
1
3
, then subtract 4.
First, multiply by Start Fraction 1 over 3 End Fraction , then subtract 4.
Answers
GPT-4o mini
To solve the equation \(3(x - 4) = 18\), one method is to first divide by 3, yielding:
\[
x - 4 = 6
\]
Then, adding 4 gives:
\[
x = 10
\]
Now, let's analyze the provided options to see which one will also lead to the same correct answer:
1. **First, distribute the 3, then add 12, and lastly divide by 3.**
- Distributing gives \(3x - 12 = 18\). Adding 12 leads to \(3x = 30\), and dividing gives \(x = 10\). (This works.)
2. **First, distribute the 3, then add 4, and lastly divide by 3.**
- Distributing gives \(3x - 12 = 18\). Adding 4 gives \(3x - 8 = 18\) which does not lead to the correct conclusion.
3. **First, divide by 3, then subtract 4.**
- Dividing gives \(x - 4 = 6\), and then subtracting 4 will not work as it would lead to \(x - 4 - 4 = 6\), which is incorrect.
4. **First, multiply by \( \frac{1}{3} \), then subtract 4.**
- Multiplying gives \(x - 4 = 6\) again. Subtracting 4 does not work since it leads to an incorrect equation.
The first option that says, "First, distribute the 3, then add 12, and lastly divide by 3," will also lead to the correct answer of \(x = 10\).
Therefore, the correct answer is:
**First, distribute the 3, then add 12, and lastly divide by 3.**
\[
x - 4 = 6
\]
Then, adding 4 gives:
\[
x = 10
\]
Now, let's analyze the provided options to see which one will also lead to the same correct answer:
1. **First, distribute the 3, then add 12, and lastly divide by 3.**
- Distributing gives \(3x - 12 = 18\). Adding 12 leads to \(3x = 30\), and dividing gives \(x = 10\). (This works.)
2. **First, distribute the 3, then add 4, and lastly divide by 3.**
- Distributing gives \(3x - 12 = 18\). Adding 4 gives \(3x - 8 = 18\) which does not lead to the correct conclusion.
3. **First, divide by 3, then subtract 4.**
- Dividing gives \(x - 4 = 6\), and then subtracting 4 will not work as it would lead to \(x - 4 - 4 = 6\), which is incorrect.
4. **First, multiply by \( \frac{1}{3} \), then subtract 4.**
- Multiplying gives \(x - 4 = 6\) again. Subtracting 4 does not work since it leads to an incorrect equation.
The first option that says, "First, distribute the 3, then add 12, and lastly divide by 3," will also lead to the correct answer of \(x = 10\).
Therefore, the correct answer is:
**First, distribute the 3, then add 12, and lastly divide by 3.**
When solving the equation β1/2(x+10)=β3
β
1
2
(
π₯
+
10
)
=
β
3
, identify the operation that should be done first.(1 point)
Responses
multiply by β2
β
2
multiply by negative 2
add 10
add 10
divide by β2
β
2
divide by negative 2
subtract 10
β
1
2
(
π₯
+
10
)
=
β
3
, identify the operation that should be done first.(1 point)
Responses
multiply by β2
β
2
multiply by negative 2
add 10
add 10
divide by β2
β
2
divide by negative 2
subtract 10
GPT-4o mini
To solve the equation \(-\frac{1}{2}(x + 10) = -3\), we start by removing the fraction. The first operation that should be done is to eliminate the negative fraction by multiplying both sides of the equation by \(-2\).
This gives:
\[
x + 10 = 6
\]
So, the operation that should be done first is:
**multiply by \(-2\)**.
This gives:
\[
x + 10 = 6
\]
So, the operation that should be done first is:
**multiply by \(-2\)**.
Three adult and three child movie tickets were purchased for $75. The cost of each child ticket is $3 less than the cost of an adult ticket. Identify the first step if solving this problem arithmetically, which would find the cost of one adult and one child ticket. (1 point)
Responses
Divide 75 by 3.
Divide 75 by 3.
Subtract 6 from 75.
Subtract 6 from 75.
Add 3 to 75.
Add 3 to 75.
Divide 75 by 6.
Responses
Divide 75 by 3.
Divide 75 by 3.
Subtract 6 from 75.
Subtract 6 from 75.
Add 3 to 75.
Add 3 to 75.
Divide 75 by 6.
GPT-4o mini
To solve this problem, we can define variables for the costs of the tickets:
- Let \( A \) be the cost of one adult ticket.
- Let \( C \) be the cost of one child ticket.
According to the information given:
1. **From the problem:** We know that
\[
C = A - 3
\]
(the cost of a child ticket is $3 less than the cost of an adult ticket).
2. **Total cost of tickets:** The total cost for 3 adult tickets and 3 child tickets is given as $75. Therefore, we can write the equation:
\[
3A + 3C = 75
\]
**Substituting the expression for \( C \)** into this equation gives:
\[
3A + 3(A - 3) = 75
\]
Now we can simplify this equation:
\[
3A + 3A - 9 = 75
\]
\[
6A - 9 = 75
\]
To isolate \( A \), we should add 9 to both sides of the equation:
\[
6A = 84
\]
Finally, we divide both sides by 6 to find the cost of one adult ticket:
\[
A = 14
\]
Now we can find the first step in solving the problem arithmetically. None of the options you've provided alternate approaches based on the derived equations. However, the correct first steps would be solving \( 3A + 3C = 75 \) directly rather than using the provided options.
None of the options listed, such as dividing 75 by 3, subtracting 6 from 75, adding 3 to 75, or dividing 75 by 6, would directly be the arithmetic step to find the costs efficiently.
From purely the context of problem-solving perspective, the first logical arithmetic step is to express the cost relationship in terms of one variable which isnβt listed. However, if forced to choose from the options as they somewhat contribute toward solving for total costs:
The most relevant would be:
**Divide 75 by 6.**
It gives a sense of averaging across the adult and child tickets although it's not a direct solving path, it hints at an average cost breakdown once replaced with valid variables respecting the given relationships.
- Let \( A \) be the cost of one adult ticket.
- Let \( C \) be the cost of one child ticket.
According to the information given:
1. **From the problem:** We know that
\[
C = A - 3
\]
(the cost of a child ticket is $3 less than the cost of an adult ticket).
2. **Total cost of tickets:** The total cost for 3 adult tickets and 3 child tickets is given as $75. Therefore, we can write the equation:
\[
3A + 3C = 75
\]
**Substituting the expression for \( C \)** into this equation gives:
\[
3A + 3(A - 3) = 75
\]
Now we can simplify this equation:
\[
3A + 3A - 9 = 75
\]
\[
6A - 9 = 75
\]
To isolate \( A \), we should add 9 to both sides of the equation:
\[
6A = 84
\]
Finally, we divide both sides by 6 to find the cost of one adult ticket:
\[
A = 14
\]
Now we can find the first step in solving the problem arithmetically. None of the options you've provided alternate approaches based on the derived equations. However, the correct first steps would be solving \( 3A + 3C = 75 \) directly rather than using the provided options.
None of the options listed, such as dividing 75 by 3, subtracting 6 from 75, adding 3 to 75, or dividing 75 by 6, would directly be the arithmetic step to find the costs efficiently.
From purely the context of problem-solving perspective, the first logical arithmetic step is to express the cost relationship in terms of one variable which isnβt listed. However, if forced to choose from the options as they somewhat contribute toward solving for total costs:
The most relevant would be:
**Divide 75 by 6.**
It gives a sense of averaging across the adult and child tickets although it's not a direct solving path, it hints at an average cost breakdown once replaced with valid variables respecting the given relationships.
Three adult and three child movie tickets were purchased for $75. The cost of each child ticket is $3 less than the cost of an adult ticket. The first step to solving this problem algebraically is to write an equation: 3(x+xβ3)=75
3
(
π₯
+
π₯
β
3
)
=
75
, where x
π₯
represents the cost of an adult ticket. Identify the next step, which would simplify the equation before solving for the variable. (1 point)
Responses
Add 3 to 75
Add 3 to 75
Divide 75 by 3.
Divide 75 by 3.
Distribute values over the variables.
Distribute values over the variables.
Combine like terms in parentheses.
3
(
π₯
+
π₯
β
3
)
=
75
, where x
π₯
represents the cost of an adult ticket. Identify the next step, which would simplify the equation before solving for the variable. (1 point)
Responses
Add 3 to 75
Add 3 to 75
Divide 75 by 3.
Divide 75 by 3.
Distribute values over the variables.
Distribute values over the variables.
Combine like terms in parentheses.
GPT-4o mini
Given the equation \(3(x + (x - 3)) = 75\), where \(x\) represents the cost of an adult ticket, the next step to simplify the equation before solving for the variable is:
**Combine like terms in parentheses.**
Let's break it down:
1. Inside the parentheses, you have \(x + (x - 3)\).
2. Combining the like terms gives you:
\[
x + x - 3 = 2x - 3
\]
So the equation becomes:
\[
3(2x - 3) = 75
\]
After this step, you can continue by distributing \(3\) over the terms and then solve for \(x\).
Hence, the correct answer is to **combine like terms in parentheses**.
**Combine like terms in parentheses.**
Let's break it down:
1. Inside the parentheses, you have \(x + (x - 3)\).
2. Combining the like terms gives you:
\[
x + x - 3 = 2x - 3
\]
So the equation becomes:
\[
3(2x - 3) = 75
\]
After this step, you can continue by distributing \(3\) over the terms and then solve for \(x\).
Hence, the correct answer is to **combine like terms in parentheses**.
Three adult and three child movie tickets were purchased for $75. The cost of each child ticket is $3 less than the cost of an adult ticket. Find the price of each ticket.(1 point)
Responses
an adult ticket is $11 and a child ticket is $8
an adult ticket is $11 and a child ticket is $8
an adult ticket is $14 and a child ticket is $11
an adult ticket is $14 and a child ticket is $11
an adult ticket is $12.50 and a child ticket is $9.50
an adult ticket is $12.50 and a child ticket is $9.50
an adult ticket is $8 and a child ticket is $5
Responses
an adult ticket is $11 and a child ticket is $8
an adult ticket is $11 and a child ticket is $8
an adult ticket is $14 and a child ticket is $11
an adult ticket is $14 and a child ticket is $11
an adult ticket is $12.50 and a child ticket is $9.50
an adult ticket is $12.50 and a child ticket is $9.50
an adult ticket is $8 and a child ticket is $5
GPT-4o mini
Let \( x \) be the cost of one adult ticket. According to the problem, the cost of one child ticket is \( x - 3 \).
We can set up the equation based on the total cost for three adult tickets and three child tickets:
\[
3x + 3(x - 3) = 75
\]
Now, simplify the equation:
1. Distribute the \( 3 \) in the second term:
\[
3x + 3x - 9 = 75
\]
2. Combine like terms:
\[
6x - 9 = 75
\]
3. Add \( 9 \) to both sides:
\[
6x = 84
\]
4. Divide by \( 6 \):
\[
x = 14
\]
Now that we have the cost of an adult ticket, we can find the cost of a child ticket:
\[
\text{Child ticket cost} = x - 3 = 14 - 3 = 11
\]
Thus, the price of each ticket is:
- An adult ticket is **$14**.
- A child ticket is **$11**.
So the correct answer is:
**an adult ticket is $14 and a child ticket is $11.**
We can set up the equation based on the total cost for three adult tickets and three child tickets:
\[
3x + 3(x - 3) = 75
\]
Now, simplify the equation:
1. Distribute the \( 3 \) in the second term:
\[
3x + 3x - 9 = 75
\]
2. Combine like terms:
\[
6x - 9 = 75
\]
3. Add \( 9 \) to both sides:
\[
6x = 84
\]
4. Divide by \( 6 \):
\[
x = 14
\]
Now that we have the cost of an adult ticket, we can find the cost of a child ticket:
\[
\text{Child ticket cost} = x - 3 = 14 - 3 = 11
\]
Thus, the price of each ticket is:
- An adult ticket is **$14**.
- A child ticket is **$11**.
So the correct answer is:
**an adult ticket is $14 and a child ticket is $11.**