Question
Kevin is four years older than his sister, June. The sum of their ages is 14. How old is Kevin?
Answers
GPT-4o mini
Let Kevin's age be \( K \) and June's age be \( J \). According to the problem, we have the following two equations:
1. \( K = J + 4 \) (Kevin is four years older than June)
2. \( K + J = 14 \) (The sum of their ages is 14)
Now we can substitute the expression for \( K \) from the first equation into the second equation:
\[
(J + 4) + J = 14
\]
This simplifies to:
\[
2J + 4 = 14
\]
Next, subtract 4 from both sides:
\[
2J = 10
\]
Now, divide both sides by 2:
\[
J = 5
\]
Now that we know June's age, we can find Kevin's age using the first equation:
\[
K = J + 4 = 5 + 4 = 9
\]
Thus, Kevin is \( \boxed{9} \) years old.
1. \( K = J + 4 \) (Kevin is four years older than June)
2. \( K + J = 14 \) (The sum of their ages is 14)
Now we can substitute the expression for \( K \) from the first equation into the second equation:
\[
(J + 4) + J = 14
\]
This simplifies to:
\[
2J + 4 = 14
\]
Next, subtract 4 from both sides:
\[
2J = 10
\]
Now, divide both sides by 2:
\[
J = 5
\]
Now that we know June's age, we can find Kevin's age using the first equation:
\[
K = J + 4 = 5 + 4 = 9
\]
Thus, Kevin is \( \boxed{9} \) years old.
x
Is (0,0)
a viable solution for the following system? Write either 1 for viable or 2 for nonviable as your response.
2y≥−3x−2
2y<−x+2
a viable solution for the following system? Write either 1 for viable or 2 for nonviable as your response.
2y≥−3x−2
2y<−x+2
GPT-4o mini
To determine if \( (0, 0) \) is a viable solution for the system of inequalities, we need to substitute \( x = 0 \) and \( y = 0 \) into both inequalities.
1. **For the first inequality**:
\[
2y \geq -3x - 2
\]
Substituting \( x = 0 \) and \( y = 0 \):
\[
2(0) \geq -3(0) - 2 \implies 0 \geq -2
\]
This statement is true.
2. **For the second inequality**:
\[
2y < -x + 2
\]
Substituting \( x = 0 \) and \( y = 0 \):
\[
2(0) < -0 + 2 \implies 0 < 2
\]
This statement is also true.
Since both inequalities are satisfied by the point \( (0, 0) \), it is a viable solution.
Thus, the answer is \( \boxed{1} \) for viable.
1. **For the first inequality**:
\[
2y \geq -3x - 2
\]
Substituting \( x = 0 \) and \( y = 0 \):
\[
2(0) \geq -3(0) - 2 \implies 0 \geq -2
\]
This statement is true.
2. **For the second inequality**:
\[
2y < -x + 2
\]
Substituting \( x = 0 \) and \( y = 0 \):
\[
2(0) < -0 + 2 \implies 0 < 2
\]
This statement is also true.
Since both inequalities are satisfied by the point \( (0, 0) \), it is a viable solution.
Thus, the answer is \( \boxed{1} \) for viable.
x
A vendor sells fruit at a farmer’s market. In one day, they sold a total of 175 oranges and pears. They sold oranges for $1.50 each and pears for $2 each, making a total of $305 dollars. If x represents the number of oranges sold and y represents the number of pears sold, is (90,85)
a viable solution to the system? Write either 1 for viable or 2 for nonviable as your response.
a viable solution to the system? Write either 1 for viable or 2 for nonviable as your response.
GPT-4o mini
To determine if \( (90, 85) \) is a viable solution for the given system, we can substitute \( x = 90 \) (the number of oranges) and \( y = 85 \) (the number of pears) into the equations based on the information provided:
1. **Total number of fruits (oranges + pears)**:
\[
x + y = 175
\]
2. **Total sales from oranges and pears**:
\[
1.50x + 2y = 305
\]
Now, let's substitute \( x = 90 \) and \( y = 85 \) into the first equation:
\[
90 + 85 = 175
\]
\[
175 = 175 \quad \text{(True)}
\]
Now, let's substitute \( x = 90 \) and \( y = 85 \) into the second equation:
\[
1.50(90) + 2(85) = 305
\]
Calculating each term:
\[
135 + 170 = 305
\]
\[
305 = 305 \quad \text{(True)}
\]
Both equations are satisfied by the point \( (90, 85) \).
Thus, \( (90, 85) \) is a viable solution.
The answer is \( \boxed{1} \) for viable.
1. **Total number of fruits (oranges + pears)**:
\[
x + y = 175
\]
2. **Total sales from oranges and pears**:
\[
1.50x + 2y = 305
\]
Now, let's substitute \( x = 90 \) and \( y = 85 \) into the first equation:
\[
90 + 85 = 175
\]
\[
175 = 175 \quad \text{(True)}
\]
Now, let's substitute \( x = 90 \) and \( y = 85 \) into the second equation:
\[
1.50(90) + 2(85) = 305
\]
Calculating each term:
\[
135 + 170 = 305
\]
\[
305 = 305 \quad \text{(True)}
\]
Both equations are satisfied by the point \( (90, 85) \).
Thus, \( (90, 85) \) is a viable solution.
The answer is \( \boxed{1} \) for viable.
x
Amy needs to purchase pencils and notebooks for her students. The number of pencils needed is at least twice as many as the number of notebooks. The pencils cost $0.25 each and the notebooks cost $4. Amy’s budget for these supplies allows for a maximum cost of $100. How do you represent the constraints of this scenario as a system of inequalities, where p is the number of pencils and n is the number of notebooks Amy will buy?(1 point)
Responses
p≥n+2
0.25p+4n≤100
p greater-than-or-equal-to n plus 2 0.25 p plus 4 n less-than-or-equal-to 100
p≥2n
0.25p+4n≤100
p greater-than-or-equal-to 2 n 0.25 p plus 4 n less-than-or-equal-to 100
p<2n
0.25p+4n≤100
p less than 2 n 0.25 p plus 4 n less-than-or-equal-to 100
p≥2n
0.25p+4n≥100
Responses
p≥n+2
0.25p+4n≤100
p greater-than-or-equal-to n plus 2 0.25 p plus 4 n less-than-or-equal-to 100
p≥2n
0.25p+4n≤100
p greater-than-or-equal-to 2 n 0.25 p plus 4 n less-than-or-equal-to 100
p<2n
0.25p+4n≤100
p less than 2 n 0.25 p plus 4 n less-than-or-equal-to 100
p≥2n
0.25p+4n≥100
GPT-4o mini
To represent the constraints of Amy's scenario as a system of inequalities where \( p \) is the number of pencils and \( n \) is the number of notebooks, we need to examine the two conditions described:
1. **The number of pencils needed is at least twice as many as the number of notebooks.** This can be represented by the inequality:
\[
p \geq 2n
\]
2. **The total cost of the pencils and notebooks should not exceed her budget of $100.** The cost of pencils is $0.25 each and notebooks is $4 each. This gives the inequality:
\[
0.25p + 4n \leq 100
\]
Now we can combine these two inequalities into a system of inequalities:
\[
\begin{cases}
p \geq 2n \\
0.25p + 4n \leq 100
\end{cases}
\]
From the choices given, the correct representation of the constraints is:
**Option**:
**p ≥ 2n**
**0.25p + 4n ≤ 100**
Thus, the answer is:
**p greater-than-or-equal-to 2 n**
**0.25 p plus 4 n less-than-or-equal-to 100**
So the selection corresponds to the second option you provided.
1. **The number of pencils needed is at least twice as many as the number of notebooks.** This can be represented by the inequality:
\[
p \geq 2n
\]
2. **The total cost of the pencils and notebooks should not exceed her budget of $100.** The cost of pencils is $0.25 each and notebooks is $4 each. This gives the inequality:
\[
0.25p + 4n \leq 100
\]
Now we can combine these two inequalities into a system of inequalities:
\[
\begin{cases}
p \geq 2n \\
0.25p + 4n \leq 100
\end{cases}
\]
From the choices given, the correct representation of the constraints is:
**Option**:
**p ≥ 2n**
**0.25p + 4n ≤ 100**
Thus, the answer is:
**p greater-than-or-equal-to 2 n**
**0.25 p plus 4 n less-than-or-equal-to 100**
So the selection corresponds to the second option you provided.
x
Adam has at least $25 in his wallet and no more than $375 in his savings. How can Adam represent this scenario as a system of linear inequalities, where w is the amount of money in his wallet and s is the amount of money in his savings?(1 point)
Responses
w≥25
s≤375
w greater-than-or-equal-to 25 s less-than-or-eqaul-to 375
w≥25
s>375
w greater-than-or-equal-to 25 s greater than 375
w>25
s≤375
w greater than 25 s less-than-or-eqaul-to 375
w≥25
s<375
w greater-than-or-equal-to 25 s less than 375
Responses
w≥25
s≤375
w greater-than-or-equal-to 25 s less-than-or-eqaul-to 375
w≥25
s>375
w greater-than-or-equal-to 25 s greater than 375
w>25
s≤375
w greater than 25 s less-than-or-eqaul-to 375
w≥25
s<375
w greater-than-or-equal-to 25 s less than 375
GPT-4o mini
To represent Adam's financial situation as a system of linear inequalities, we need to consider the constraints given in the problem:
1. Adam has **at least $25 in his wallet**. This means the amount in his wallet \( w \) must be greater than or equal to 25, which can be represented as:
\[
w \geq 25
\]
2. Adam has **no more than $375 in his savings**. This means the amount in his savings \( s \) must be less than or equal to 375, which can be represented as:
\[
s \leq 375
\]
Putting these two inequalities together, we have the system:
\[
\begin{cases}
w \geq 25 \\
s \leq 375
\end{cases}
\]
From the options provided, the correct representation is:
**Option**:
**w ≥ 25**
**s ≤ 375**
This corresponds to the response:
**w greater-than-or-equal-to 25 s less-than-or-equal-to 375**.
Thus, the final answer is:
**w greater-than-or-equal-to 25 s less-than-or-equal-to 375**.
1. Adam has **at least $25 in his wallet**. This means the amount in his wallet \( w \) must be greater than or equal to 25, which can be represented as:
\[
w \geq 25
\]
2. Adam has **no more than $375 in his savings**. This means the amount in his savings \( s \) must be less than or equal to 375, which can be represented as:
\[
s \leq 375
\]
Putting these two inequalities together, we have the system:
\[
\begin{cases}
w \geq 25 \\
s \leq 375
\end{cases}
\]
From the options provided, the correct representation is:
**Option**:
**w ≥ 25**
**s ≤ 375**
This corresponds to the response:
**w greater-than-or-equal-to 25 s less-than-or-equal-to 375**.
Thus, the final answer is:
**w greater-than-or-equal-to 25 s less-than-or-equal-to 375**.
x
Is (0,−3)
a viable solution for the following system?
y<−5x+2
3y>3x−6
(1 point)
Responses
(0,−3)
is a viable solution since it makes both inequalities true.
left parenthesis 0 comma negative 3 right parenthesis is a viable solution since it makes both inequalities true.
(0,−3)
is a viable solution since it makes one inequality true.
left parenthesis 0 comma negative 3 right parenthesis is a viable solution since it makes one inequality true.
(0,−3)
is a nonviable solution since it makes both inequalities true.
left parenthesis 0 comma negative 3 right parenthesis is a nonviable solution since it makes both inequalities true.
(0,−3)
is a nonviable solution since it does not make both inequalities true.
a viable solution for the following system?
y<−5x+2
3y>3x−6
(1 point)
Responses
(0,−3)
is a viable solution since it makes both inequalities true.
left parenthesis 0 comma negative 3 right parenthesis is a viable solution since it makes both inequalities true.
(0,−3)
is a viable solution since it makes one inequality true.
left parenthesis 0 comma negative 3 right parenthesis is a viable solution since it makes one inequality true.
(0,−3)
is a nonviable solution since it makes both inequalities true.
left parenthesis 0 comma negative 3 right parenthesis is a nonviable solution since it makes both inequalities true.
(0,−3)
is a nonviable solution since it does not make both inequalities true.
GPT-4o mini
To determine if \( (0, -3) \) is a viable solution for the given system of inequalities, we will substitute \( x = 0 \) and \( y = -3 \) into both inequalities:
1. **First inequality**:
\[
y < -5x + 2
\]
Substituting \( x = 0 \) and \( y = -3 \):
\[
-3 < -5(0) + 2
\]
This simplifies to:
\[
-3 < 2 \quad \text{(True)}
\]
2. **Second inequality**:
\[
3y > 3x - 6
\]
Substituting \( x = 0 \) and \( y = -3 \):
\[
3(-3) > 3(0) - 6
\]
This simplifies to:
\[
-9 > -6 \quad \text{(False)}
\]
Now, since the first inequality is true and the second inequality is false, \( (0, -3) \) does not satisfy both inequalities.
Thus, the conclusion is:
**(0,−3) is a nonviable solution since it does not make both inequalities true.**
The correct response is:
**(0,−3) is a nonviable solution since it does not make both inequalities true.**
1. **First inequality**:
\[
y < -5x + 2
\]
Substituting \( x = 0 \) and \( y = -3 \):
\[
-3 < -5(0) + 2
\]
This simplifies to:
\[
-3 < 2 \quad \text{(True)}
\]
2. **Second inequality**:
\[
3y > 3x - 6
\]
Substituting \( x = 0 \) and \( y = -3 \):
\[
3(-3) > 3(0) - 6
\]
This simplifies to:
\[
-9 > -6 \quad \text{(False)}
\]
Now, since the first inequality is true and the second inequality is false, \( (0, -3) \) does not satisfy both inequalities.
Thus, the conclusion is:
**(0,−3) is a nonviable solution since it does not make both inequalities true.**
The correct response is:
**(0,−3) is a nonviable solution since it does not make both inequalities true.**