Question
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≥2n
0.25p+4n≥100
p greater-than-or-equal-to 2 n 0.25 p plus 4 n greater-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≥n+2
0.25p+4n≤100
Responses
p≥2n
0.25p+4n≥100
p greater-than-or-equal-to 2 n 0.25 p plus 4 n greater-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≥n+2
0.25p+4n≤100
Answers
GPT-4o mini
To represent the constraints of Amy's situation as a system of inequalities, we need to consider the two main conditions given:
1. The number of pencils needed (p) is at least twice the number of notebooks (n), which can be represented by the inequality:
\[
p \geq 2n
\]
2. The total cost of the pencils and notebooks must not exceed Amy’s budget of $100. The cost of pencils is $0.25 each and the cost of notebooks is $4 each. Therefore, this constraint can be represented by the inequality:
\[
0.25p + 4n \leq 100
\]
Considering these two inequalities together, the correct representation of the constraints is:
\[
p \geq 2n
\]
\[
0.25p + 4n \leq 100
\]
So the final response that accurately reflects this is:
**p≥2n**
**0.25p+4n≤100**
1. The number of pencils needed (p) is at least twice the number of notebooks (n), which can be represented by the inequality:
\[
p \geq 2n
\]
2. The total cost of the pencils and notebooks must not exceed Amy’s budget of $100. The cost of pencils is $0.25 each and the cost of notebooks is $4 each. Therefore, this constraint can be represented by the inequality:
\[
0.25p + 4n \leq 100
\]
Considering these two inequalities together, the correct representation of the constraints is:
\[
p \geq 2n
\]
\[
0.25p + 4n \leq 100
\]
So the final response that accurately reflects this is:
**p≥2n**
**0.25p+4n≤100**
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 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-or-equal-to 25 s less than 375
w≥25
s≤375
Responses
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 greater than 375
w≥25
s<375
w greater-than-or-equal-to 25 s less than 375
w≥25
s≤375
GPT-4o mini
To represent Adam's financial situation as a system of linear inequalities, we can use the following variables:
- \( w \) represents the amount of money in Adam's wallet.
- \( s \) represents the amount of money in his savings.
Given the conditions:
1. Adam has at least $25 in his wallet, which can be represented as:
\[
w \geq 25
\]
2. Adam has no more than $375 in his savings, which can be represented as:
\[
s \leq 375
\]
So, the system of inequalities that represents this scenario is:
\[
w \geq 25
\]
\[
s \leq 375
\]
The correct response that captures this representation is:
**w≥25**
**s≤375**
- \( w \) represents the amount of money in Adam's wallet.
- \( s \) represents the amount of money in his savings.
Given the conditions:
1. Adam has at least $25 in his wallet, which can be represented as:
\[
w \geq 25
\]
2. Adam has no more than $375 in his savings, which can be represented as:
\[
s \leq 375
\]
So, the system of inequalities that represents this scenario is:
\[
w \geq 25
\]
\[
s \leq 375
\]
The correct response that captures this representation is:
**w≥25**
**s≤375**
Is (0,−3) a viable solution for the following system?
y<−5x+2
3y>3x−6
(1 point)
Responses
(0,−3) is a nonviable solution since it does not make both inequalities true.
left parenthesis 0 comma negative 3 right parenthesis is a nonviable solution since it does not make both inequalities 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 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.
y<−5x+2
3y>3x−6
(1 point)
Responses
(0,−3) is a nonviable solution since it does not make both inequalities true.
left parenthesis 0 comma negative 3 right parenthesis is a nonviable solution since it does not make both inequalities 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 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.
GPT-4o mini
To determine whether \((0, -3)\) is a viable solution for the system of inequalities given, we need to evaluate it in both inequalities:
1. **For the first inequality:**
\[
y < -5x + 2
\]
Substituting \(x = 0\) and \(y = -3\):
\[
-3 < -5(0) + 2 \implies -3 < 2
\]
This inequality is true.
2. **For the second inequality:**
\[
3y > 3x - 6
\]
Again substituting \(x = 0\) and \(y = -3\):
\[
3(-3) > 3(0) - 6 \implies -9 > -6
\]
This inequality is false.
Since \((0, -3)\) satisfies the first inequality but does not satisfy the second inequality, it cannot be considered a viable solution for the system of inequalities as both inequalities must hold true.
Therefore, the correct response is:
**(0,−3) is a nonviable solution since it does not make both inequalities true.**
1. **For the first inequality:**
\[
y < -5x + 2
\]
Substituting \(x = 0\) and \(y = -3\):
\[
-3 < -5(0) + 2 \implies -3 < 2
\]
This inequality is true.
2. **For the second inequality:**
\[
3y > 3x - 6
\]
Again substituting \(x = 0\) and \(y = -3\):
\[
3(-3) > 3(0) - 6 \implies -9 > -6
\]
This inequality is false.
Since \((0, -3)\) satisfies the first inequality but does not satisfy the second inequality, it cannot be considered a viable solution for the system of inequalities as both inequalities must hold true.
Therefore, the correct response is:
**(0,−3) is a nonviable solution since it does not make both inequalities true.**
Is (−2,2) a viable solution for the following system? Explain.
−7x+2y=18
3x+3y=0
(1 point)
Responses
(−2,2) is a viable solution since it makes both equations true.
left parenthesis negative 2 comma 2 right parenthesis is a viable solution since it makes both equations true.
(−2,2) is a nonviable solution since it makes neither equation true.
left parenthesis negative 2 comma 2 right parenthesis is a nonviable solution since it makes neither equation true.
(−2,2) is a nonviable solution since it makes one equation true.
left parenthesis negative 2 comma 2 right parenthesis is a nonviable solution since it makes one equation true.
(−2,2) is a viable solution since it makes one equation true.
left parenthesis negative 2 comma 2 right parenthesis is a viable solution since it makes one equation true.
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−7x+2y=18
3x+3y=0
(1 point)
Responses
(−2,2) is a viable solution since it makes both equations true.
left parenthesis negative 2 comma 2 right parenthesis is a viable solution since it makes both equations true.
(−2,2) is a nonviable solution since it makes neither equation true.
left parenthesis negative 2 comma 2 right parenthesis is a nonviable solution since it makes neither equation true.
(−2,2) is a nonviable solution since it makes one equation true.
left parenthesis negative 2 comma 2 right parenthesis is a nonviable solution since it makes one equation true.
(−2,2) is a viable solution since it makes one equation true.
left parenthesis negative 2 comma 2 right parenthesis is a viable solution since it makes one equation true.
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GPT-4o mini
To determine whether \((-2, 2)\) is a viable solution for the system of equations, we need to substitute \(x = -2\) and \(y = 2\) into both equations:
1. **For the first equation:**
\[
-7x + 2y = 18
\]
Substituting \(x = -2\) and \(y = 2\):
\[
-7(-2) + 2(2) = 18 \\
14 + 4 = 18 \\
18 = 18
\]
This equation is true.
2. **For the second equation:**
\[
3x + 3y = 0
\]
Substituting \(x = -2\) and \(y = 2\):
\[
3(-2) + 3(2) = 0 \\
-6 + 6 = 0 \\
0 = 0
\]
This equation is also true.
Since \((-2, 2)\) satisfies both equations, it is a viable solution.
The correct response is:
**(−2,2) is a viable solution since it makes both equations true.**
1. **For the first equation:**
\[
-7x + 2y = 18
\]
Substituting \(x = -2\) and \(y = 2\):
\[
-7(-2) + 2(2) = 18 \\
14 + 4 = 18 \\
18 = 18
\]
This equation is true.
2. **For the second equation:**
\[
3x + 3y = 0
\]
Substituting \(x = -2\) and \(y = 2\):
\[
3(-2) + 3(2) = 0 \\
-6 + 6 = 0 \\
0 = 0
\]
This equation is also true.
Since \((-2, 2)\) satisfies both equations, it is a viable solution.
The correct response is:
**(−2,2) is a viable solution since it makes both equations true.**
The sum of twice the number of green marbles and the number of yellow marbles in a bag is more than 20. The total number of green and yellow marbles is less than 15. Which of the following is a viable solution to the system of inequalities, where x is the number of green marbles and y is the number of yellow marbles?(1 point)
Responses
(6,7)
left parenthesis 6 comma 7 right parenthesis
(10,5)
left parenthesis 10 comma 5 right parenthesis
(3,12)
left parenthesis 3 comma 12 right parenthesis
(8,6)
Responses
(6,7)
left parenthesis 6 comma 7 right parenthesis
(10,5)
left parenthesis 10 comma 5 right parenthesis
(3,12)
left parenthesis 3 comma 12 right parenthesis
(8,6)
GPT-4o mini
To determine which pair of values is a viable solution to the system of inequalities based on the given conditions, we first need to translate the problem into inequalities:
1. The sum of twice the number of green marbles and the number of yellow marbles is more than 20:
\[
2x + y > 20
\]
2. The total number of green and yellow marbles is less than 15:
\[
x + y < 15
\]
Next, we will test each option provided to see if it satisfies both inequalities.
1. **For (6, 7):**
- \(2x + y = 2(6) + 7 = 12 + 7 = 19\) (not > 20)
- \(x + y = 6 + 7 = 13\) (< 15)
- **Result:** Does not satisfy the first inequality.
2. **For (10, 5):**
- \(2x + y = 2(10) + 5 = 20 + 5 = 25\) (> 20)
- \(x + y = 10 + 5 = 15\) (not < 15)
- **Result:** Does not satisfy the second inequality.
3. **For (3, 12):**
- \(2x + y = 2(3) + 12 = 6 + 12 = 18\) (not > 20)
- \(x + y = 3 + 12 = 15\) (not < 15)
- **Result:** Does not satisfy either inequality.
4. **For (8, 6):**
- \(2x + y = 2(8) + 6 = 16 + 6 = 22\) (> 20)
- \(x + y = 8 + 6 = 14\) (< 15)
- **Result:** Satisfies both inequalities.
Since (8, 6) satisfies both inequalities, it is the viable solution.
Thus, the correct response is:
**(8, 6)**
1. The sum of twice the number of green marbles and the number of yellow marbles is more than 20:
\[
2x + y > 20
\]
2. The total number of green and yellow marbles is less than 15:
\[
x + y < 15
\]
Next, we will test each option provided to see if it satisfies both inequalities.
1. **For (6, 7):**
- \(2x + y = 2(6) + 7 = 12 + 7 = 19\) (not > 20)
- \(x + y = 6 + 7 = 13\) (< 15)
- **Result:** Does not satisfy the first inequality.
2. **For (10, 5):**
- \(2x + y = 2(10) + 5 = 20 + 5 = 25\) (> 20)
- \(x + y = 10 + 5 = 15\) (not < 15)
- **Result:** Does not satisfy the second inequality.
3. **For (3, 12):**
- \(2x + y = 2(3) + 12 = 6 + 12 = 18\) (not > 20)
- \(x + y = 3 + 12 = 15\) (not < 15)
- **Result:** Does not satisfy either inequality.
4. **For (8, 6):**
- \(2x + y = 2(8) + 6 = 16 + 6 = 22\) (> 20)
- \(x + y = 8 + 6 = 14\) (< 15)
- **Result:** Satisfies both inequalities.
Since (8, 6) satisfies both inequalities, it is the viable solution.
Thus, the correct response is:
**(8, 6)**