Question
Given the table for x, sweater knitted, and y, money earned:
x: # of sweaters 7 6 2 9 3
y: $ earned 392 336 112 504 168
(4 points)
a. Every sweater knitted is (BLANK) dollars earned.
b. Dollars/1 sweater is called a (BLANK).
c. This relationship is (BLANK) because 0 sweaters knitted is
dollars earned.
x: # of sweaters 7 6 2 9 3
y: $ earned 392 336 112 504 168
(4 points)
a. Every sweater knitted is (BLANK) dollars earned.
b. Dollars/1 sweater is called a (BLANK).
c. This relationship is (BLANK) because 0 sweaters knitted is
dollars earned.
Answers
GPT-4o mini
Let's analyze the data you've provided:
### Given data:
- **x (Number of Sweaters)**: 7, 6, 2, 9, 3
- **y (Dollars Earned)**: 392, 336, 112, 504, 168
To calculate the earnings per sweater, we can find the earnings by dividing the total dollars earned by the number of sweaters knitted for each entry.
### Calculating the earnings per sweater:
1. For 7 sweaters: $$392 / 7 = 56$$
2. For 6 sweaters: $$336 / 6 = 56$$
3. For 2 sweaters: $$112 / 2 = 56$$
4. For 9 sweaters: $$504 / 9 = 56$$
5. For 3 sweaters: $$168 / 3 = 56$$
Every entry shows that the earnings per sweater is consistently $56.
### Answers to the blanks:
a. Every sweater knitted is **56** dollars earned.
b. Dollars/1 sweater is called a **unit price** (or **price per unit**).
c. This relationship is **linear** because 0 sweaters knitted is **0** dollars earned.
So the completed sentences are:
- a. Every sweater knitted is **56** dollars earned.
- b. Dollars/1 sweater is called a **unit price**.
- c. This relationship is **linear** because 0 sweaters knitted is **0** dollars earned.
### Given data:
- **x (Number of Sweaters)**: 7, 6, 2, 9, 3
- **y (Dollars Earned)**: 392, 336, 112, 504, 168
To calculate the earnings per sweater, we can find the earnings by dividing the total dollars earned by the number of sweaters knitted for each entry.
### Calculating the earnings per sweater:
1. For 7 sweaters: $$392 / 7 = 56$$
2. For 6 sweaters: $$336 / 6 = 56$$
3. For 2 sweaters: $$112 / 2 = 56$$
4. For 9 sweaters: $$504 / 9 = 56$$
5. For 3 sweaters: $$168 / 3 = 56$$
Every entry shows that the earnings per sweater is consistently $56.
### Answers to the blanks:
a. Every sweater knitted is **56** dollars earned.
b. Dollars/1 sweater is called a **unit price** (or **price per unit**).
c. This relationship is **linear** because 0 sweaters knitted is **0** dollars earned.
So the completed sentences are:
- a. Every sweater knitted is **56** dollars earned.
- b. Dollars/1 sweater is called a **unit price**.
- c. This relationship is **linear** because 0 sweaters knitted is **0** dollars earned.
Given the table for x, time in minutes, and y, distance traveled:
x: time 8 5 7 2 9
y: distance 664 415 581 166 747
(4 points)
a. Every minute
meters are traveled.
b. Write an equation to show the relationship between x and y:
.
c. This relationship is
because the y-intercept is
.
Question 5
5. A cab company charges $12 per mile for a lift to the airport.
What change would the company make to their charges to make this a non proportional situation?
(1 point)
Responses
Charge $15 per mile instead of $12
Charge $15 per mile instead of $12
Charge a flat rate of $20 and then $12 per mile
Charge a flat rate of $20 and then $12 per mile
Charge $4 per mile instead of $12
Charge $4 per mile instead of $12
No changes are needed.
No changes are needed.
Question 6
6. One business charges $22 per t-shirt that is custom made.
Another business charges $16 per t-shirt plus a $15 set up fee for custom made t-shirts.
How would you transform the first graph to make it become a non proportional graph with the same y-intercept as the second graph?
(1 point)
Responses
Down 4
Down 4
Left 15
Left 15
Up 15
Up 15
Up 4
Up 4
Down 15
Down 15
Right 4
x: time 8 5 7 2 9
y: distance 664 415 581 166 747
(4 points)
a. Every minute
meters are traveled.
b. Write an equation to show the relationship between x and y:
.
c. This relationship is
because the y-intercept is
.
Question 5
5. A cab company charges $12 per mile for a lift to the airport.
What change would the company make to their charges to make this a non proportional situation?
(1 point)
Responses
Charge $15 per mile instead of $12
Charge $15 per mile instead of $12
Charge a flat rate of $20 and then $12 per mile
Charge a flat rate of $20 and then $12 per mile
Charge $4 per mile instead of $12
Charge $4 per mile instead of $12
No changes are needed.
No changes are needed.
Question 6
6. One business charges $22 per t-shirt that is custom made.
Another business charges $16 per t-shirt plus a $15 set up fee for custom made t-shirts.
How would you transform the first graph to make it become a non proportional graph with the same y-intercept as the second graph?
(1 point)
Responses
Down 4
Down 4
Left 15
Left 15
Up 15
Up 15
Up 4
Up 4
Down 15
Down 15
Right 4
GPT-4o mini
Let's analyze the data provided and answer the questions step by step.
### Given Data:
- **x (time in minutes)**: 8, 5, 7, 2, 9
- **y (distance traveled in meters)**: 664, 415, 581, 166, 747
### Step 1: Calculate Distance per Minute
To find out how many meters are traveled per minute, we can calculate the rate of distance per minute by dividing the distance traveled (y) by the time (x) for each entry.
1. For \( x = 8 \): \( 664 / 8 = 83.0 \)
2. For \( x = 5 \): \( 415 / 5 = 83.0 \)
3. For \( x = 7 \): \( 581 / 7 = 83.0 \)
4. For \( x = 2 \): \( 166 / 2 = 83.0 \)
5. For \( x = 9 \): \( 747 / 9 = 83.0 \)
Every minute, **83.0 meters** are traveled.
### Step 2: Write an Equation
The relationship between \( x \) and \( y \) can be expressed as a direct proportionality equation. The general linear equation form is:
\[ y = mx + b \]
where \( m \) is the rate (slope) and \( b \) is the y-intercept.
From the calculations, we know the slope \( m \) (distance per minute) is 83. Since there is no time (x = 0), the distance would be 0, meaning the y-intercept \( b \) is 0.
Thus, the equation is:
\[ y = 83x \]
### Step 3: Relationship Characterization
This relationship is **linear** because the y-intercept is **0**.
### Answers to the Blanks:
a. Every minute **83** meters are traveled.
b. Write an equation to show the relationship between x and y: **y = 83x**.
c. This relationship is **linear** because the y-intercept is **0**.
---
### Question 5: Cab Company Charges
To make the situation a non-proportional one, the company needs to add a fixed charge (like a base fee) that does not depend on the distance traveled. Therefore, the correct choice is:
- **Charge a flat rate of $20 and then $12 per mile.**
This adds a fixed cost (the $20) which makes the total charge not directly proportional to the number of miles traveled.
---
### Question 6: Transforming the First Graph
The first business charges $22 per t-shirt and is a proportional relationship, while the second business charges $16 per t-shirt plus a $15 setup fee, making it a non-proportional relationship (with a y-intercept of 15).
To transform the first graph (which has a y-intercept of 0) so that its y-intercept matches that of the second graph (which is 15), you would need to move the graph up by 15 units.
Thus, the correct response is:
- **Up 15.**
This would create a situation where the new intercept of the first graph matches the intercept of the second graph, making them both non-proportional.
### Given Data:
- **x (time in minutes)**: 8, 5, 7, 2, 9
- **y (distance traveled in meters)**: 664, 415, 581, 166, 747
### Step 1: Calculate Distance per Minute
To find out how many meters are traveled per minute, we can calculate the rate of distance per minute by dividing the distance traveled (y) by the time (x) for each entry.
1. For \( x = 8 \): \( 664 / 8 = 83.0 \)
2. For \( x = 5 \): \( 415 / 5 = 83.0 \)
3. For \( x = 7 \): \( 581 / 7 = 83.0 \)
4. For \( x = 2 \): \( 166 / 2 = 83.0 \)
5. For \( x = 9 \): \( 747 / 9 = 83.0 \)
Every minute, **83.0 meters** are traveled.
### Step 2: Write an Equation
The relationship between \( x \) and \( y \) can be expressed as a direct proportionality equation. The general linear equation form is:
\[ y = mx + b \]
where \( m \) is the rate (slope) and \( b \) is the y-intercept.
From the calculations, we know the slope \( m \) (distance per minute) is 83. Since there is no time (x = 0), the distance would be 0, meaning the y-intercept \( b \) is 0.
Thus, the equation is:
\[ y = 83x \]
### Step 3: Relationship Characterization
This relationship is **linear** because the y-intercept is **0**.
### Answers to the Blanks:
a. Every minute **83** meters are traveled.
b. Write an equation to show the relationship between x and y: **y = 83x**.
c. This relationship is **linear** because the y-intercept is **0**.
---
### Question 5: Cab Company Charges
To make the situation a non-proportional one, the company needs to add a fixed charge (like a base fee) that does not depend on the distance traveled. Therefore, the correct choice is:
- **Charge a flat rate of $20 and then $12 per mile.**
This adds a fixed cost (the $20) which makes the total charge not directly proportional to the number of miles traveled.
---
### Question 6: Transforming the First Graph
The first business charges $22 per t-shirt and is a proportional relationship, while the second business charges $16 per t-shirt plus a $15 setup fee, making it a non-proportional relationship (with a y-intercept of 15).
To transform the first graph (which has a y-intercept of 0) so that its y-intercept matches that of the second graph (which is 15), you would need to move the graph up by 15 units.
Thus, the correct response is:
- **Up 15.**
This would create a situation where the new intercept of the first graph matches the intercept of the second graph, making them both non-proportional.
Solve this equation for y: 7y=12x (1 point)
Responses
y=7/12xy is equal to 7 over 12 x
y=12/7xy is equal to 12 sevenths x
y=12x−7y is equal to 12 x minus 7
y=12x+7y is equal to 12 x plus 7
Question 2
8. Change the equation from standard form to slope-intercept form by solving for y.
8x−4y=24
Determine who changed the equation correctly:
Bob:
Subtract 8x from both sides: −4y=−8x+24
Divide by -4 on both sides: y=2x−6
Bill:
Subtract 8x from both sides: −4y=16x
Divide by -4 on both sides: y=−4x
Bud:
Subtract 8x from both sides: −4y=−8x+24
Divide by -4 on both sides: y=2x+24
(1 point)
Responses
Bill
Bill
Bud
Bud
None of them are correct
None of them are correct
Bob
Bob
Question 3
9. Solve 3x+2b=6c for x.
Step 1: 3x+2b−2b=6c−2b subtract 2b from both sides
Step 2: 3x=6c−2b combine like terms/simplify
Step 3: 3x/3=6c/3−2b divide by 3 on both sides to get x isolated
Step 4: x=2c−2b simplify/final answer
Which statement is TRUE?
(1 point)
Responses
The process is correct.
The process is correct.
The process has an error. The correct answer is x=−4b−cThe process has an error. The correct answer is x is equal to negative 4 b minus c
The process has an error. The correct answer is x=2c−23bThe process has an error. The correct answer is x is equal to 2 c minus 2 thirds b
The process has an error. The correct answer is x=4bc3The process has an error. The correct answer is x is equal to the fraction with numerator 4 b c and denominator 3
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Responses
y=7/12xy is equal to 7 over 12 x
y=12/7xy is equal to 12 sevenths x
y=12x−7y is equal to 12 x minus 7
y=12x+7y is equal to 12 x plus 7
Question 2
8. Change the equation from standard form to slope-intercept form by solving for y.
8x−4y=24
Determine who changed the equation correctly:
Bob:
Subtract 8x from both sides: −4y=−8x+24
Divide by -4 on both sides: y=2x−6
Bill:
Subtract 8x from both sides: −4y=16x
Divide by -4 on both sides: y=−4x
Bud:
Subtract 8x from both sides: −4y=−8x+24
Divide by -4 on both sides: y=2x+24
(1 point)
Responses
Bill
Bill
Bud
Bud
None of them are correct
None of them are correct
Bob
Bob
Question 3
9. Solve 3x+2b=6c for x.
Step 1: 3x+2b−2b=6c−2b subtract 2b from both sides
Step 2: 3x=6c−2b combine like terms/simplify
Step 3: 3x/3=6c/3−2b divide by 3 on both sides to get x isolated
Step 4: x=2c−2b simplify/final answer
Which statement is TRUE?
(1 point)
Responses
The process is correct.
The process is correct.
The process has an error. The correct answer is x=−4b−cThe process has an error. The correct answer is x is equal to negative 4 b minus c
The process has an error. The correct answer is x=2c−23bThe process has an error. The correct answer is x is equal to 2 c minus 2 thirds b
The process has an error. The correct answer is x=4bc3The process has an error. The correct answer is x is equal to the fraction with numerator 4 b c and denominator 3
Skip to navigation
GPT-4o mini
Let's address each question one by one:
### Question 1: Solve the equation for y
**Equation:** \( 7y = 12x \)
To solve for \( y \), divide both sides by 7:
\[
y = \frac{12}{7}x
\]
**Responses:**
- y = \( \frac{12}{7} \)x **(Correct)**
### Question 2: Change the equation from standard form to slope-intercept form
**Equation:** \( 8x - 4y = 24 \)
1. **Subtract \( 8x \):**
\(-4y = -8x + 24\)
2. **Divide by \(-4\):**
\[
y = 2x - 6
\]
**Determine who changed the equation correctly:**
- **Bob's solution is correct:** \( y = 2x - 6 \)
- Bill has made an error in stating \( -4y = 16x \).
- Bud also made an error in stating \( y = 2x + 24 \).
**Response:** Bob
### Question 3: Solve for x in the equation
**Equation:** \( 3x + 2b = 6c \)
1. **Step 1:**
\[
3x + 2b - 2b = 6c - 2b \Rightarrow 3x = 6c - 2b
\]
2. **Step 2:**
Combine terms (done correctly):
\[
3x = 6c - 2b
\]
3. **Step 3:**
Divide both sides by 3:
\[
x = \frac{6c - 2b}{3}
\]
4. **Step 4:**
Simplify:
\[
x = 2c - \frac{2b}{3}
\]
**Evaluate the process:**
- The process is mostly correct until the final answer where the simplification is not expressed properly. The true transformation should yield:
\[
x = 2c - \frac{2b}{3}
\]
The given statement options for the "correct answer":
- The process does not have an exact representation in the options provided. However, it appears that **none of the listed true final formulations** accurately state \( x = 2c - \frac{2b}{3} \).
**Responses:**
- The process is correct. (but with a note of the final simplification being slightly misrepresented)
- The process has an error. The correct answer is \( x = 2c - \frac{2}{3}b \) **(This option aligns most closely with the correct representation)**.
Therefore, **Responses:**
1. **Question 1:** y = \( \frac{12}{7} \)x
2. **Question 2:** Bob
3. **Question 3:** The process is correct. (with an explanation about the simplification noted)
### Question 1: Solve the equation for y
**Equation:** \( 7y = 12x \)
To solve for \( y \), divide both sides by 7:
\[
y = \frac{12}{7}x
\]
**Responses:**
- y = \( \frac{12}{7} \)x **(Correct)**
### Question 2: Change the equation from standard form to slope-intercept form
**Equation:** \( 8x - 4y = 24 \)
1. **Subtract \( 8x \):**
\(-4y = -8x + 24\)
2. **Divide by \(-4\):**
\[
y = 2x - 6
\]
**Determine who changed the equation correctly:**
- **Bob's solution is correct:** \( y = 2x - 6 \)
- Bill has made an error in stating \( -4y = 16x \).
- Bud also made an error in stating \( y = 2x + 24 \).
**Response:** Bob
### Question 3: Solve for x in the equation
**Equation:** \( 3x + 2b = 6c \)
1. **Step 1:**
\[
3x + 2b - 2b = 6c - 2b \Rightarrow 3x = 6c - 2b
\]
2. **Step 2:**
Combine terms (done correctly):
\[
3x = 6c - 2b
\]
3. **Step 3:**
Divide both sides by 3:
\[
x = \frac{6c - 2b}{3}
\]
4. **Step 4:**
Simplify:
\[
x = 2c - \frac{2b}{3}
\]
**Evaluate the process:**
- The process is mostly correct until the final answer where the simplification is not expressed properly. The true transformation should yield:
\[
x = 2c - \frac{2b}{3}
\]
The given statement options for the "correct answer":
- The process does not have an exact representation in the options provided. However, it appears that **none of the listed true final formulations** accurately state \( x = 2c - \frac{2b}{3} \).
**Responses:**
- The process is correct. (but with a note of the final simplification being slightly misrepresented)
- The process has an error. The correct answer is \( x = 2c - \frac{2}{3}b \) **(This option aligns most closely with the correct representation)**.
Therefore, **Responses:**
1. **Question 1:** y = \( \frac{12}{7} \)x
2. **Question 2:** Bob
3. **Question 3:** The process is correct. (with an explanation about the simplification noted)
A graph is shown with a negative slope with three dots
Dot 1: (6, 1)
Dot 2: (3, 3)
Dot 3: (0, 5)
What is the y-intercept? Response area
b. What is the slope? Response area
c. What is the equation of the line in slope-intercept form? Response area
d. This is a (Proportional/non-proportional) relationship.
Dot 1: (6, 1)
Dot 2: (3, 3)
Dot 3: (0, 5)
What is the y-intercept? Response area
b. What is the slope? Response area
c. What is the equation of the line in slope-intercept form? Response area
d. This is a (Proportional/non-proportional) relationship.
GPT-4o mini
Let's analyze the information provided about the graph and its points:
### Given Points:
1. Dot 1: \( (6, 1) \)
2. Dot 2: \( (3, 3) \)
3. Dot 3: \( (0, 5) \)
### a. What is the y-intercept?
The y-intercept is the value of \( y \) when \( x = 0 \). From the given points, we see that:
- Dot 3: \( (0, 5) \)
So, the **y-intercept** is **5**.
### b. What is the slope?
The slope (\( m \)) of a line can be calculated using two points on the line with the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using Dot 1 \( (6, 1) \) and Dot 2 \( (3, 3) \):
\[
m = \frac{3 - 1}{3 - 6} = \frac{2}{-3} = -\frac{2}{3}
\]
Thus, the **slope** is **-\(\frac{2}{3}\)**.
### c. What is the equation of the line in slope-intercept form?
The slope-intercept form of a line is given by the equation:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
Using the slope of \(-\frac{2}{3}\) and the y-intercept of \(5\):
\[
y = -\frac{2}{3}x + 5
\]
So, the equation of the line in slope-intercept form is:
\[
y = -\frac{2}{3}x + 5
\]
### d. Is this a Proportional or Non-proportional relationship?
A proportional relationship must pass through the origin \( (0, 0) \) and be in the form \( y = mx \). Since the y-intercept is \(5\) (not \(0\)), this indicates that it does not pass through the origin.
Thus, this is a **Non-proportional** relationship.
### Summary of Answers:
a. The y-intercept is **5**.
b. The slope is **-\(\frac{2}{3}\)**.
c. The equation of the line in slope-intercept form is **\(y = -\frac{2}{3}x + 5\)**.
d. This is a **Non-proportional** relationship.
### Given Points:
1. Dot 1: \( (6, 1) \)
2. Dot 2: \( (3, 3) \)
3. Dot 3: \( (0, 5) \)
### a. What is the y-intercept?
The y-intercept is the value of \( y \) when \( x = 0 \). From the given points, we see that:
- Dot 3: \( (0, 5) \)
So, the **y-intercept** is **5**.
### b. What is the slope?
The slope (\( m \)) of a line can be calculated using two points on the line with the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using Dot 1 \( (6, 1) \) and Dot 2 \( (3, 3) \):
\[
m = \frac{3 - 1}{3 - 6} = \frac{2}{-3} = -\frac{2}{3}
\]
Thus, the **slope** is **-\(\frac{2}{3}\)**.
### c. What is the equation of the line in slope-intercept form?
The slope-intercept form of a line is given by the equation:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
Using the slope of \(-\frac{2}{3}\) and the y-intercept of \(5\):
\[
y = -\frac{2}{3}x + 5
\]
So, the equation of the line in slope-intercept form is:
\[
y = -\frac{2}{3}x + 5
\]
### d. Is this a Proportional or Non-proportional relationship?
A proportional relationship must pass through the origin \( (0, 0) \) and be in the form \( y = mx \). Since the y-intercept is \(5\) (not \(0\)), this indicates that it does not pass through the origin.
Thus, this is a **Non-proportional** relationship.
### Summary of Answers:
a. The y-intercept is **5**.
b. The slope is **-\(\frac{2}{3}\)**.
c. The equation of the line in slope-intercept form is **\(y = -\frac{2}{3}x + 5\)**.
d. This is a **Non-proportional** relationship.
Suppose that the water level of a river is 45 feet after 3 days of no rain and 10 feet after 10 days of no rain. The river was 60 feet before the drought began.(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
a. What is the rate of change (slope) for this scenario? Response area
b. What is the y-intercept for this scenario? Response area
c. Write an equation for the scenario in slope-intercept form. Response area
d. Write an equation in standard form. Response area
Question 2
14. Suppose that the water level of a river is 45 feet after 3 days of no rain and 10 feet after 10 days of no rain. Given the two points we know: (3, 45) and (10, 10), what would the point slope equation look like?(1 point)
Responses
(y−3)=−5(x−45)open paren y minus 3 close paren is equal to negative 5 times open paren x minus 45 close paren
(y+10)=−5(x−10)open paren y plus 10 close paren is equal to negative 5 times open paren x minus 10 close paren
(y−10)=−5(x+10)open paren y minus 10 close paren is equal to negative 5 times open paren x plus 10 close paren
(y−45)=−5(x−3)open paren y minus 45 close paren is equal to negative 5 times open paren x minus 3 close paren
Question 3
15. Which graph matches the linear equation: −2x+3y=−6(1 point)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
a. What is the rate of change (slope) for this scenario? Response area
b. What is the y-intercept for this scenario? Response area
c. Write an equation for the scenario in slope-intercept form. Response area
d. Write an equation in standard form. Response area
Question 2
14. Suppose that the water level of a river is 45 feet after 3 days of no rain and 10 feet after 10 days of no rain. Given the two points we know: (3, 45) and (10, 10), what would the point slope equation look like?(1 point)
Responses
(y−3)=−5(x−45)open paren y minus 3 close paren is equal to negative 5 times open paren x minus 45 close paren
(y+10)=−5(x−10)open paren y plus 10 close paren is equal to negative 5 times open paren x minus 10 close paren
(y−10)=−5(x+10)open paren y minus 10 close paren is equal to negative 5 times open paren x plus 10 close paren
(y−45)=−5(x−3)open paren y minus 45 close paren is equal to negative 5 times open paren x minus 3 close paren
Question 3
15. Which graph matches the linear equation: −2x+3y=−6(1 point)
GPT-4o mini
Let’s solve the given problems step by step:
### Given Information:
- Water level after 3 days (Point 1): \( (3, 45) \)
- Water level after 10 days (Point 2): \( (10, 10) \)
- Initial water level before the drought began: 60 feet (which is not explicitly used for the calculations here but provides context).
### Question Responses:
#### a. What is the rate of change (slope) for this scenario?
The slope (\( m \)) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \( (3, 45) \) and \( (10, 10) \):
\[
m = \frac{10 - 45}{10 - 3} = \frac{-35}{7} = -5
\]
Thus, the **rate of change (slope)** is **-5**.
#### b. What is the y-intercept for this scenario?
To find the y-intercept (\( b \)), we can use the slope-intercept form of the equation \( y = mx + b \). We can take one of the points, for example, \( (3, 45) \), and substitute \( m \) and the coordinates to solve for \( b \):
\[
45 = -5(3) + b
\]
\[
45 = -15 + b
\]
\[
b = 45 + 15 = 60
\]
Thus, the **y-intercept** is **60**.
#### c. Write an equation for the scenario in slope-intercept form.
Using the slope \( -5 \) and y-intercept \( 60 \):
\[
y = -5x + 60
\]
#### d. Write an equation in standard form.
To convert the slope-intercept form \( y = -5x + 60 \) to standard form \( Ax + By = C\):
1. Rearrange to standard form:
\[
5x + y = 60
\]
So the **equation in standard form** is **\( 5x + y = 60 \)**.
### Summary of Responses:
a. The rate of change (slope) is **-5**.
b. The y-intercept is **60**.
c. The equation in slope-intercept form is **\( y = -5x + 60 \)**.
d. The equation in standard form is **\( 5x + y = 60 \)**.
---
### Question 2: What would the point slope equation look like?
Given the slope from earlier \( -5 \) and taking the point (3, 45), the point-slope form of the equation is:
\[
y - y_1 = m(x - x_1)
\]
Substituting for \( m = -5 \), \( x_1 = 3 \), and \( y_1 = 45\):
\[
y - 45 = -5(x - 3)
\]
This form matches the form:
- **(y − 45) = −5(x − 3)** (the fourth option you've provided).
### Question 3: Which graph matches the linear equation: \(-2x + 3y = -6\)?
To determine which graph matches the equation \(-2x + 3y = -6\), let’s convert this to slope-intercept form:
1. Rearrange in slope-intercept form:
\[
3y = 2x - 6
\]
\[
y = \frac{2}{3}x - 2
\]
This indicates that the slope is \( \frac{2}{3} \) and the y-intercept is \(-2\). The graph should have a positive slope and cross the y-axis at y = -2.
**Unfortunately, I can't visualize the graphs provided**, but you should look for the graph that has a slope of \( \frac{2}{3} \) and passes through the point (0, -2) on the y-axis.
### Summary of Question 2 and 3 Answers:
**Question 2:** \( (y - 45) = -5(x - 3) \).
**Question 3:** Look for the graph that has a slope of \( \frac{2}{3} \) and passes through the point (0, -2) on the y-axis.
### Given Information:
- Water level after 3 days (Point 1): \( (3, 45) \)
- Water level after 10 days (Point 2): \( (10, 10) \)
- Initial water level before the drought began: 60 feet (which is not explicitly used for the calculations here but provides context).
### Question Responses:
#### a. What is the rate of change (slope) for this scenario?
The slope (\( m \)) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \( (3, 45) \) and \( (10, 10) \):
\[
m = \frac{10 - 45}{10 - 3} = \frac{-35}{7} = -5
\]
Thus, the **rate of change (slope)** is **-5**.
#### b. What is the y-intercept for this scenario?
To find the y-intercept (\( b \)), we can use the slope-intercept form of the equation \( y = mx + b \). We can take one of the points, for example, \( (3, 45) \), and substitute \( m \) and the coordinates to solve for \( b \):
\[
45 = -5(3) + b
\]
\[
45 = -15 + b
\]
\[
b = 45 + 15 = 60
\]
Thus, the **y-intercept** is **60**.
#### c. Write an equation for the scenario in slope-intercept form.
Using the slope \( -5 \) and y-intercept \( 60 \):
\[
y = -5x + 60
\]
#### d. Write an equation in standard form.
To convert the slope-intercept form \( y = -5x + 60 \) to standard form \( Ax + By = C\):
1. Rearrange to standard form:
\[
5x + y = 60
\]
So the **equation in standard form** is **\( 5x + y = 60 \)**.
### Summary of Responses:
a. The rate of change (slope) is **-5**.
b. The y-intercept is **60**.
c. The equation in slope-intercept form is **\( y = -5x + 60 \)**.
d. The equation in standard form is **\( 5x + y = 60 \)**.
---
### Question 2: What would the point slope equation look like?
Given the slope from earlier \( -5 \) and taking the point (3, 45), the point-slope form of the equation is:
\[
y - y_1 = m(x - x_1)
\]
Substituting for \( m = -5 \), \( x_1 = 3 \), and \( y_1 = 45\):
\[
y - 45 = -5(x - 3)
\]
This form matches the form:
- **(y − 45) = −5(x − 3)** (the fourth option you've provided).
### Question 3: Which graph matches the linear equation: \(-2x + 3y = -6\)?
To determine which graph matches the equation \(-2x + 3y = -6\), let’s convert this to slope-intercept form:
1. Rearrange in slope-intercept form:
\[
3y = 2x - 6
\]
\[
y = \frac{2}{3}x - 2
\]
This indicates that the slope is \( \frac{2}{3} \) and the y-intercept is \(-2\). The graph should have a positive slope and cross the y-axis at y = -2.
**Unfortunately, I can't visualize the graphs provided**, but you should look for the graph that has a slope of \( \frac{2}{3} \) and passes through the point (0, -2) on the y-axis.
### Summary of Question 2 and 3 Answers:
**Question 2:** \( (y - 45) = -5(x - 3) \).
**Question 3:** Look for the graph that has a slope of \( \frac{2}{3} \) and passes through the point (0, -2) on the y-axis.
The tables below show the number of jumping jacks completed after a given period of time in seconds.
Kimberly:
Time (seconds) Jumping Jacks
3 17
8 37
12 53
16 69
Katrina:
Time (seconds) Jumping Jacks
2 10
5 25
12 60
20 100(3 points)
a. Which person is doing more jumping jacks per second?
b. Which person had done more jumping jacks initially before the timer started?
c. Which person shows a proportional relationship?
Kimberly:
Time (seconds) Jumping Jacks
3 17
8 37
12 53
16 69
Katrina:
Time (seconds) Jumping Jacks
2 10
5 25
12 60
20 100(3 points)
a. Which person is doing more jumping jacks per second?
b. Which person had done more jumping jacks initially before the timer started?
c. Which person shows a proportional relationship?
GPT-4o mini
To analyze the jumping jacks completed by Kimberly and Katrina, we will first calculate the rate of jumping jacks per second for each individual and look at the initial values.
### Data:
**Kimberly:**
- Time (seconds): 3, 8, 12, 16
- Jumping Jacks: 17, 37, 53, 69
**Katrina:**
- Time (seconds): 2, 5, 12, 20
- Jumping Jacks: 10, 25, 60, 100
### a. Which person is doing more jumping jacks per second?
To calculate the rate of jumping jacks per second, we will divide the number of jumping jacks by the corresponding time in seconds for each person.
**Kimberly:**
1. For 3 seconds: \( \frac{17}{3} \approx 5.67 \) jacks/second
2. For 8 seconds: \( \frac{37}{8} = 4.625 \) jacks/second
3. For 12 seconds: \( \frac{53}{12} \approx 4.42 \) jacks/second
4. For 16 seconds: \( \frac{69}{16} \approx 4.31 \) jacks/second
Average rate for Kimberly:
\[
\text{Average} = \frac{5.67 + 4.625 + 4.42 + 4.31}{4} \approx 4.77 \text{ jacks/second}
\]
**Katrina:**
1. For 2 seconds: \( \frac{10}{2} = 5 \) jacks/second
2. For 5 seconds: \( \frac{25}{5} = 5 \) jacks/second
3. For 12 seconds: \( \frac{60}{12} = 5 \) jacks/second
4. For 20 seconds: \( \frac{100}{20} = 5 \) jacks/second
Average rate for Katrina:
\[
\text{Average} = \frac{5 + 5 + 5 + 5}{4} = 5 \text{ jacks/second}
\]
**Conclusion for Part a:**
Katrina is doing more jumping jacks per second with an average of **5** jacks/second compared to Kimberly's average of **4.77** jacks/second.
### b. Which person had done more jumping jacks initially before the timer started?
To analyze the initial number of jumping jacks, we need to look at the values at \(t = 0\) (if explicitly given): however, both tables do not provide the initial value directly but can be inferred by examining the first data points since we only have data from 2 seconds or more.
- For Kimberly, the lowest time is 3 seconds with 17 jacks.
- For Katrina, the lowest time is 2 seconds with 10 jacks.
**Conclusion for Part b:**
Since both have a recorded jumping jack count after starting timers, we can't conclude who had them at \(t=0\), but both have numbers higher than zero at their minimum recorded time. Without explicit zero-time values, we can't confirm, but the hint could imply Katrina was lower initially than Kimberly at the start of their counts.
### c. Which person shows a proportional relationship?
A proportional relationship means the ratio of jumping jacks to time remains constant.
**Kimberly:**
- The rates vary (in decimal approximations from above), meaning Kimberly's result is **not proportional**.
**Katrina:**
- The rate is constant at \(5\) jacks/second throughout all time measurements, indicating that Katrina's relationship is **proportional**.
**Conclusion for Part c:**
Katrina shows a **proportional relationship**.
### Final Summary:
a. **Katrina** is doing more jumping jacks per second.
b. It can't be definitively concluded who had more initially due to lack of explicit \(t=0\) data.
c. **Katrina** shows a proportional relationship.
### Data:
**Kimberly:**
- Time (seconds): 3, 8, 12, 16
- Jumping Jacks: 17, 37, 53, 69
**Katrina:**
- Time (seconds): 2, 5, 12, 20
- Jumping Jacks: 10, 25, 60, 100
### a. Which person is doing more jumping jacks per second?
To calculate the rate of jumping jacks per second, we will divide the number of jumping jacks by the corresponding time in seconds for each person.
**Kimberly:**
1. For 3 seconds: \( \frac{17}{3} \approx 5.67 \) jacks/second
2. For 8 seconds: \( \frac{37}{8} = 4.625 \) jacks/second
3. For 12 seconds: \( \frac{53}{12} \approx 4.42 \) jacks/second
4. For 16 seconds: \( \frac{69}{16} \approx 4.31 \) jacks/second
Average rate for Kimberly:
\[
\text{Average} = \frac{5.67 + 4.625 + 4.42 + 4.31}{4} \approx 4.77 \text{ jacks/second}
\]
**Katrina:**
1. For 2 seconds: \( \frac{10}{2} = 5 \) jacks/second
2. For 5 seconds: \( \frac{25}{5} = 5 \) jacks/second
3. For 12 seconds: \( \frac{60}{12} = 5 \) jacks/second
4. For 20 seconds: \( \frac{100}{20} = 5 \) jacks/second
Average rate for Katrina:
\[
\text{Average} = \frac{5 + 5 + 5 + 5}{4} = 5 \text{ jacks/second}
\]
**Conclusion for Part a:**
Katrina is doing more jumping jacks per second with an average of **5** jacks/second compared to Kimberly's average of **4.77** jacks/second.
### b. Which person had done more jumping jacks initially before the timer started?
To analyze the initial number of jumping jacks, we need to look at the values at \(t = 0\) (if explicitly given): however, both tables do not provide the initial value directly but can be inferred by examining the first data points since we only have data from 2 seconds or more.
- For Kimberly, the lowest time is 3 seconds with 17 jacks.
- For Katrina, the lowest time is 2 seconds with 10 jacks.
**Conclusion for Part b:**
Since both have a recorded jumping jack count after starting timers, we can't conclude who had them at \(t=0\), but both have numbers higher than zero at their minimum recorded time. Without explicit zero-time values, we can't confirm, but the hint could imply Katrina was lower initially than Kimberly at the start of their counts.
### c. Which person shows a proportional relationship?
A proportional relationship means the ratio of jumping jacks to time remains constant.
**Kimberly:**
- The rates vary (in decimal approximations from above), meaning Kimberly's result is **not proportional**.
**Katrina:**
- The rate is constant at \(5\) jacks/second throughout all time measurements, indicating that Katrina's relationship is **proportional**.
**Conclusion for Part c:**
Katrina shows a **proportional relationship**.
### Final Summary:
a. **Katrina** is doing more jumping jacks per second.
b. It can't be definitively concluded who had more initially due to lack of explicit \(t=0\) data.
c. **Katrina** shows a proportional relationship.