Question
A line passes through the point (0, –1) and has a positive slope. Which of these points could that line pass through? Check all that apply.
(12, 3)
(–2, –5)
(–3, 1)
(1, 15)
(5, –2)
just to say it is not just or one of the options
(12, 3) and (1, 15) maybe they are both correct but whats the last one? or one of them arent and its another answer
(12, 3)
(–2, –5)
(–3, 1)
(1, 15)
(5, –2)
just to say it is not just or one of the options
(12, 3) and (1, 15) maybe they are both correct but whats the last one? or one of them arent and its another answer
Answers
Answered by
GPT-4o mini
To determine which of the provided points could lie on a line that passes through the point (0, -1) and has a positive slope, we can analyze the relationship between the y-coordinates of the given points and the y-coordinate of the point at x = 0 (which is -1).
A line with a positive slope will increase in y-value as x increases. Therefore, for a point (x, y) to be on the line, the y-coordinate should be greater than -1 when x is greater than 0 (to the right of the point (0, -1)) and less than -1 when x is less than 0 (to the left of the point (0, -1)).
Let's check each point:
1. **(12, 3)**: Here, x = 12 (which is greater than 0) and y = 3 (which is greater than -1). So, this point could be on the line.
2. **(–2, –5)**: Here, x = -2 (which is less than 0) and y = -5 (which is less than -1). So, this point could also be on the line.
3. **(–3, 1)**: Here, x = -3 (which is less than 0) and y = 1 (which is greater than -1). So, this point cannot be on the line.
4. **(1, 15)**: Here, x = 1 (which is greater than 0) and y = 15 (which is greater than -1). So, this point could be on the line.
5. **(5, –2)**: Here, x = 5 (which is greater than 0) and y = -2 (which is less than -1). So, this point cannot be on the line.
From our analysis, the points that the line could pass through are:
- (12, 3)
- (–2, –5)
- (1, 15)
Thus, the valid points are (12, 3), (–2, –5), and (1, 15).
A line with a positive slope will increase in y-value as x increases. Therefore, for a point (x, y) to be on the line, the y-coordinate should be greater than -1 when x is greater than 0 (to the right of the point (0, -1)) and less than -1 when x is less than 0 (to the left of the point (0, -1)).
Let's check each point:
1. **(12, 3)**: Here, x = 12 (which is greater than 0) and y = 3 (which is greater than -1). So, this point could be on the line.
2. **(–2, –5)**: Here, x = -2 (which is less than 0) and y = -5 (which is less than -1). So, this point could also be on the line.
3. **(–3, 1)**: Here, x = -3 (which is less than 0) and y = 1 (which is greater than -1). So, this point cannot be on the line.
4. **(1, 15)**: Here, x = 1 (which is greater than 0) and y = 15 (which is greater than -1). So, this point could be on the line.
5. **(5, –2)**: Here, x = 5 (which is greater than 0) and y = -2 (which is less than -1). So, this point cannot be on the line.
From our analysis, the points that the line could pass through are:
- (12, 3)
- (–2, –5)
- (1, 15)
Thus, the valid points are (12, 3), (–2, –5), and (1, 15).
Answer
Anya found the slope of the line that passes through the points (–7, 4) and (2, –3). Her work is shown below.Let (x2, y2) be (–7, 4) and (x1, y1) be (2, –3).m = m equals StartFraction x 2 minus x 1 over y 2 minus y 1. = Equals StartFraction negative 7 minus 2 Over 4 minus negative 3. = Negative StartFraction 9 Over 7 EndFraction.The slope is Negative StartFraction 9 Over 7 EndFraction..What error did she make?She simplified the denominator incorrectly. The denominator simplifies to –7.She labeled the points incorrectly. The point (–7, 4) should be (x1, y1).She used an incorrect formula. The formula should be the change in y-values with respect to the change in the x-values.She used an incorrect formula. The formula should be the sum of the x-values with respect to the sum of the y-values.
Answered by
GPT-4o mini
Let’s review Anya’s calculations to identify the error she made.
The formula for the slope \( m \) of a line that passes through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
In this case, we have the points \( (–7, 4) \) and \( (2, –3) \). We should label \( (x_1, y_1) = (2, -3) \) and \( (x_2, y_2) = (-7, 4) \).
Plugging in the coordinates into the formula, we have:
- \( x_1 = 2, y_1 = -3 \)
- \( x_2 = -7, y_2 = 4 \)
Now substituting these into the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-3)}{-7 - 2}
\]
Calculating the numerator:
\[
4 - (-3) = 4 + 3 = 7
\]
Calculating the denominator:
\[
-7 - 2 = -9
\]
So the slope \( m \) is:
\[
m = \frac{7}{-9} = -\frac{7}{9}
\]
**Now evaluating Anya's work:**
1. **She simplified the denominator incorrectly:** This is true since she claimed it simplifies to -7 when it should be -9.
2. **She labeled the points incorrectly:** Anya correctly identified \( (–7, 4) \) as \( (x_2, y_2) \), so this statement is incorrect.
3. **She used an incorrect formula (change in y-values with respect to change in x-values):** This is incorrect; she used the right formula.
4. **She used an incorrect formula (sum of the x-values with respect to the sum of the y-values):** This is not applicable here; she used the correct slope formula.
### Summary of the Error:
Anya made a mistake in the denominator, which she incorrectly simplified, hence **the correct explanation for her mistake is:**
**"She simplified the denominator incorrectly."**
The formula for the slope \( m \) of a line that passes through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
In this case, we have the points \( (–7, 4) \) and \( (2, –3) \). We should label \( (x_1, y_1) = (2, -3) \) and \( (x_2, y_2) = (-7, 4) \).
Plugging in the coordinates into the formula, we have:
- \( x_1 = 2, y_1 = -3 \)
- \( x_2 = -7, y_2 = 4 \)
Now substituting these into the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-3)}{-7 - 2}
\]
Calculating the numerator:
\[
4 - (-3) = 4 + 3 = 7
\]
Calculating the denominator:
\[
-7 - 2 = -9
\]
So the slope \( m \) is:
\[
m = \frac{7}{-9} = -\frac{7}{9}
\]
**Now evaluating Anya's work:**
1. **She simplified the denominator incorrectly:** This is true since she claimed it simplifies to -7 when it should be -9.
2. **She labeled the points incorrectly:** Anya correctly identified \( (–7, 4) \) as \( (x_2, y_2) \), so this statement is incorrect.
3. **She used an incorrect formula (change in y-values with respect to change in x-values):** This is incorrect; she used the right formula.
4. **She used an incorrect formula (sum of the x-values with respect to the sum of the y-values):** This is not applicable here; she used the correct slope formula.
### Summary of the Error:
Anya made a mistake in the denominator, which she incorrectly simplified, hence **the correct explanation for her mistake is:**
**"She simplified the denominator incorrectly."**
Answer
Not right another statement please
Answered by
GPT-4o mini
Let’s revisit Anya’s calculation step-by-step and clarify the mistake she made in her reasoning.
Anya used the points \( (–7, 4) \) and \( (2, –3) \) but labeled them incorrectly while applying the slope formula. According to her labels:
- She set \( (x_2, y_2) = (–7, 4) \)
- She set \( (x_1, y_1) = (2, –3) \)
However, in the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
she incorrectly substituted for the coordinates. The proper labeling should be:
- \( (x_1, y_1) = (2, -3) \)
- \( (x_2, y_2) = (-7, 4) \)
This means that when she computed \( y_2 - y_1 \) and \( x_2 - x_1 \), she did not correctly assign the values for \( y_1 \) and \( y_2 \) in her calculation.
Thus, the most accurate statement reflecting Anya's error is:
**"She labeled the points incorrectly. The point (–7, 4) should be (x1, y1)."**
This statement properly describes the main source of her error in determining the slope.
Anya used the points \( (–7, 4) \) and \( (2, –3) \) but labeled them incorrectly while applying the slope formula. According to her labels:
- She set \( (x_2, y_2) = (–7, 4) \)
- She set \( (x_1, y_1) = (2, –3) \)
However, in the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
she incorrectly substituted for the coordinates. The proper labeling should be:
- \( (x_1, y_1) = (2, -3) \)
- \( (x_2, y_2) = (-7, 4) \)
This means that when she computed \( y_2 - y_1 \) and \( x_2 - x_1 \), she did not correctly assign the values for \( y_1 \) and \( y_2 \) in her calculation.
Thus, the most accurate statement reflecting Anya's error is:
**"She labeled the points incorrectly. The point (–7, 4) should be (x1, y1)."**
This statement properly describes the main source of her error in determining the slope.
Answer
She used an incorrect formula. The formula should be the change in y-values with respect to the change in the x-values. wrong poopoo
Answered by
GPT-4o mini
I apologize for the confusion in my previous responses. Let’s clarify Anya's calculation and identify the exact nature of her error more accurately.
Anya's work shows that she calculated the slope using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
However, she misidentified the points.
**Correctly identifying the points:**
- For the point \( (–7, 4) \):
- Let \( (x_1, y_1) = (2, -3) \)
- Let \( (x_2, y_2) = (–7, 4) \)
So, substituting correctly:
- **y-coordinates:** \( y_2 - y_1 = 4 - (-3) = 4 + 3 = 7 \)
- **x-coordinates:** \( x_2 - x_1 = -7 - 2 = -9 \)
Thus, when she calculated the slope, it should have resulted in:
\[
m = \frac{7}{-9} = -\frac{7}{9}
\]
**Her specific error was that she did not properly label the points** while applying the slope formula, which directly affected her calculation.
The correct statement regarding her mistake is:
**"She labeled the points incorrectly. The point (–7, 4) should be (x1, y1)."**
Again, I appreciate your patience, and I'm sorry for the earlier confusion. Thank you for giving me the opportunity to clarify!
Anya's work shows that she calculated the slope using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
However, she misidentified the points.
**Correctly identifying the points:**
- For the point \( (–7, 4) \):
- Let \( (x_1, y_1) = (2, -3) \)
- Let \( (x_2, y_2) = (–7, 4) \)
So, substituting correctly:
- **y-coordinates:** \( y_2 - y_1 = 4 - (-3) = 4 + 3 = 7 \)
- **x-coordinates:** \( x_2 - x_1 = -7 - 2 = -9 \)
Thus, when she calculated the slope, it should have resulted in:
\[
m = \frac{7}{-9} = -\frac{7}{9}
\]
**Her specific error was that she did not properly label the points** while applying the slope formula, which directly affected her calculation.
The correct statement regarding her mistake is:
**"She labeled the points incorrectly. The point (–7, 4) should be (x1, y1)."**
Again, I appreciate your patience, and I'm sorry for the earlier confusion. Thank you for giving me the opportunity to clarify!