Question
The table shows values for (x) and (y). Here are the details:
| (x) | (y) |
|--------|--------|
| -14 | 8 |
| -7 | 6 |
| 0 | 4 |
| 7 | 2 |
| 14 | 0 |
This indicates a relationship where as (x) increases, (y) decreases. If you have any specific questions or need further analysis regarding this data, let me know!
Find the slope of the line that passes through the points shown in the table.
The slope of the line that passes through the points in the table is
.
| (x) | (y) |
|--------|--------|
| -14 | 8 |
| -7 | 6 |
| 0 | 4 |
| 7 | 2 |
| 14 | 0 |
This indicates a relationship where as (x) increases, (y) decreases. If you have any specific questions or need further analysis regarding this data, let me know!
Find the slope of the line that passes through the points shown in the table.
The slope of the line that passes through the points in the table is
.
Answers
Answer
To determine the type of slope for each pair of points, we can use the formula for calculating slope:
[
\text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1}
]
Let's analyze the given pairs of points:
Points (3, 5) and (-1, 2):
[
m = \frac{2 - 5}{-1 - 3} = \frac{-3}{-4} = \frac{3}{4} \quad \text{(positive slope)}
]
Points (6, -3) and (4, -3):
[
m = \frac{-3 - (-3)}{4 - 6} = \frac{0}{-2} = 0 \quad \text{(zero slope)}
]
Points (-7, 8) and (7, 0):
[
m = \frac{0 - 8}{7 - (-7)} = \frac{-8}{14} = -\frac{4}{7} \quad \text{(negative slope)}
]
Points (2, 4) and (5, 1):
[
m = \frac{1 - 4}{5 - 2} = \frac{-3}{3} = -1 \quad \text{(negative slope)}
]
So the classifications are as follows:
(3, 5) and (-1, 2): Positive slope
(6, -3) and (4, -3): Zero slope
(-7, 8) and (7, 0): Negative slope
(2, 4) and (5, 1): Negative slope
[
\text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1}
]
Let's analyze the given pairs of points:
Points (3, 5) and (-1, 2):
[
m = \frac{2 - 5}{-1 - 3} = \frac{-3}{-4} = \frac{3}{4} \quad \text{(positive slope)}
]
Points (6, -3) and (4, -3):
[
m = \frac{-3 - (-3)}{4 - 6} = \frac{0}{-2} = 0 \quad \text{(zero slope)}
]
Points (-7, 8) and (7, 0):
[
m = \frac{0 - 8}{7 - (-7)} = \frac{-8}{14} = -\frac{4}{7} \quad \text{(negative slope)}
]
Points (2, 4) and (5, 1):
[
m = \frac{1 - 4}{5 - 2} = \frac{-3}{3} = -1 \quad \text{(negative slope)}
]
So the classifications are as follows:
(3, 5) and (-1, 2): Positive slope
(6, -3) and (4, -3): Zero slope
(-7, 8) and (7, 0): Negative slope
(2, 4) and (5, 1): Negative slope
Answer
- **(-7, 8) and (7, 0)**: **Negative slope** UNDENFINED
Answer
The table shows the gallons of water in a pool over time.A two column table with six rows. The first column, Time in minutes, has the entries, 0, 1, 2, 3, 4, 5. The second column, Water in Pool in gallons, has the entries, 50, 44, 38, 32, 26, 20.Choose the term that describes the slope of the line of a graph representing the data in the table.The slope of a line graphed to represent the volume of water in a pool over time would be described as_________.
Answer
A coordinate plane with a straight line with a positive slope starting at (0, 0) and passing through the points (1.0, 10) and (2.0, 20).Use the graph to find the slope. slope = slope equals StartFraction rise Over run EndFraction. The runner’s speed is minutes per mile.
Answer
Which linear function represents a slope of ? A two column table with five rows. The first column, x, has the entries, 3, 6, 9, 12. The second column, y, has the entries, negative 11, 1, 13, 25. A coordinate plane with a straight line with a positive slope passing through (0, 3), (4, 4), and (8, 7). A two column table with five rows. The first column, x, has the entries, negative 5, negative 1, 3, 7. The second column, y, has the entries, 32, 24, 16, 8.A coordinate plane with a straight line with a positive slope passing through (2, 0), (3, 4), and (4, 8)
Answered by
GPT-4o mini
To find the slope of the line that passes through the points given in the table, we can use the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
We can select any two points from the table to calculate the slope. Let's use the first point \((-14, 8)\) and the last point \((14, 0)\):
- \( (x_1, y_1) = (-14, 8) \)
- \( (x_2, y_2) = (14, 0) \)
Now, we substitute these values into the slope formula:
\[
m = \frac{0 - 8}{14 - (-14)} = \frac{-8}{14 + 14} = \frac{-8}{28} = \frac{-2}{7}
\]
Therefore, the slope of the line that passes through the points shown in the table is
\[
\frac{-2}{7}.
\]
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
We can select any two points from the table to calculate the slope. Let's use the first point \((-14, 8)\) and the last point \((14, 0)\):
- \( (x_1, y_1) = (-14, 8) \)
- \( (x_2, y_2) = (14, 0) \)
Now, we substitute these values into the slope formula:
\[
m = \frac{0 - 8}{14 - (-14)} = \frac{-8}{14 + 14} = \frac{-8}{28} = \frac{-2}{7}
\]
Therefore, the slope of the line that passes through the points shown in the table is
\[
\frac{-2}{7}.
\]
Answered by
GPT-4o mini
Your analysis of the slopes for the pairs of points is correct. Let's summarize the results clearly:
1. **Points (3, 5) and (-1, 2)**:
\[
m = \frac{2 - 5}{-1 - 3} = \frac{-3}{-4} = \frac{3}{4} \quad \text{(positive slope)}
\]
2. **Points (6, -3) and (4, -3)**:
\[
m = \frac{-3 - (-3)}{4 - 6} = \frac{0}{-2} = 0 \quad \text{(zero slope)}
\]
3. **Points (-7, 8) and (7, 0)**:
\[
m = \frac{0 - 8}{7 - (-7)} = \frac{-8}{14} = -\frac{4}{7} \quad \text{(negative slope)}
\]
4. **Points (2, 4) and (5, 1)**:
\[
m = \frac{1 - 4}{5 - 2} = \frac{-3}{3} = -1 \quad \text{(negative slope)}
\]
### Summary of Classifications:
- **(3, 5) and (-1, 2)**: **Positive slope**
- **(6, -3) and (4, -3)**: **Zero slope**
- **(-7, 8) and (7, 0)**: **Negative slope**
- **(2, 4) and (5, 1)**: **Negative slope**
Your classification is accurate, and you have correctly calculated the slopes for each pair of points. If you have any further questions or need additional analysis, feel free to ask!
1. **Points (3, 5) and (-1, 2)**:
\[
m = \frac{2 - 5}{-1 - 3} = \frac{-3}{-4} = \frac{3}{4} \quad \text{(positive slope)}
\]
2. **Points (6, -3) and (4, -3)**:
\[
m = \frac{-3 - (-3)}{4 - 6} = \frac{0}{-2} = 0 \quad \text{(zero slope)}
\]
3. **Points (-7, 8) and (7, 0)**:
\[
m = \frac{0 - 8}{7 - (-7)} = \frac{-8}{14} = -\frac{4}{7} \quad \text{(negative slope)}
\]
4. **Points (2, 4) and (5, 1)**:
\[
m = \frac{1 - 4}{5 - 2} = \frac{-3}{3} = -1 \quad \text{(negative slope)}
\]
### Summary of Classifications:
- **(3, 5) and (-1, 2)**: **Positive slope**
- **(6, -3) and (4, -3)**: **Zero slope**
- **(-7, 8) and (7, 0)**: **Negative slope**
- **(2, 4) and (5, 1)**: **Negative slope**
Your classification is accurate, and you have correctly calculated the slopes for each pair of points. If you have any further questions or need additional analysis, feel free to ask!
Answered by
GPT-4o mini
Let's revisit the calculation for the slope between the points \((-7, 8)\) and \((7, 0)\):
The slope \(m\) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where:
- \((x_1, y_1) = (-7, 8)\)
- \((x_2, y_2) = (7, 0)\)
Substituting these values:
\[
m = \frac{0 - 8}{7 - (-7)} = \frac{-8}{7 + 7} = \frac{-8}{14} = -\frac{4}{7}
\]
### Correction:
The calculation here is indeed correct, and the slope is \(-\frac{4}{7}\), which is a negative slope—not undefined.
A slope will only be undefined if the line is vertical (where the change in \(x\) is zero, resulting in a division by zero). In this case, since \(x_1 = -7\) and \(x_2 = 7\), the change in \(x\) is \(14\), so the slope is defined.
### Conclusion:
- The slope between the points \((-7, 8)\) and \((7, 0)\) is \(-\frac{4}{7}\), which is indeed a **negative slope**.
If there was any misunderstanding or if you meant a different pair of points, please clarify!
The slope \(m\) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where:
- \((x_1, y_1) = (-7, 8)\)
- \((x_2, y_2) = (7, 0)\)
Substituting these values:
\[
m = \frac{0 - 8}{7 - (-7)} = \frac{-8}{7 + 7} = \frac{-8}{14} = -\frac{4}{7}
\]
### Correction:
The calculation here is indeed correct, and the slope is \(-\frac{4}{7}\), which is a negative slope—not undefined.
A slope will only be undefined if the line is vertical (where the change in \(x\) is zero, resulting in a division by zero). In this case, since \(x_1 = -7\) and \(x_2 = 7\), the change in \(x\) is \(14\), so the slope is defined.
### Conclusion:
- The slope between the points \((-7, 8)\) and \((7, 0)\) is \(-\frac{4}{7}\), which is indeed a **negative slope**.
If there was any misunderstanding or if you meant a different pair of points, please clarify!
Answered by
GPT-4o mini
To describe the slope of the line representing the volume of water in the pool over time, let's first analyze the data given in the table:
| Time (minutes) | Water in Pool (gallons) |
|----------------|----------------------------|
| 0 | 50 |
| 1 | 44 |
| 2 | 38 |
| 3 | 32 |
| 4 | 26 |
| 5 | 20 |
### Observations:
- As time increases from 0 to 5 minutes, the amount of water in the pool decreases from 50 gallons to 20 gallons.
### Calculation of Slope:
Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), let's consider two points to calculate the slope. We can take the first point (0, 50) and the last point (5, 20):
- Point 1: \((x_1, y_1) = (0, 50)\)
- Point 2: \((x_2, y_2) = (5, 20)\)
Now substituting in the slope formula:
\[
m = \frac{20 - 50}{5 - 0} = \frac{-30}{5} = -6
\]
### Conclusion:
The slope is negative, which means that as time goes on, the amount of water in the pool decreases at a rate of 6 gallons per minute.
Thus, the appropriate term to fill in the blank:
The slope of a line graphed to represent the volume of water in a pool over time would be described as **negative**.
| Time (minutes) | Water in Pool (gallons) |
|----------------|----------------------------|
| 0 | 50 |
| 1 | 44 |
| 2 | 38 |
| 3 | 32 |
| 4 | 26 |
| 5 | 20 |
### Observations:
- As time increases from 0 to 5 minutes, the amount of water in the pool decreases from 50 gallons to 20 gallons.
### Calculation of Slope:
Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), let's consider two points to calculate the slope. We can take the first point (0, 50) and the last point (5, 20):
- Point 1: \((x_1, y_1) = (0, 50)\)
- Point 2: \((x_2, y_2) = (5, 20)\)
Now substituting in the slope formula:
\[
m = \frac{20 - 50}{5 - 0} = \frac{-30}{5} = -6
\]
### Conclusion:
The slope is negative, which means that as time goes on, the amount of water in the pool decreases at a rate of 6 gallons per minute.
Thus, the appropriate term to fill in the blank:
The slope of a line graphed to represent the volume of water in a pool over time would be described as **negative**.
Answered by
GPT-4o mini
To find the slope of a line using the points you provided, we can use the slope formula:
\[
\text{slope} (m) = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}
\]
You mentioned two points: \((1.0, 10)\) and \((2.0, 20)\). Let's assign these points:
- Point 1: \((x_1, y_1) = (1.0, 10)\)
- Point 2: \((x_2, y_2) = (2.0, 20)\)
Now substituting into the slope formula:
\[
m = \frac{20 - 10}{2.0 - 1.0} = \frac{10}{1.0} = 10
\]
So, the slope of the line is \(10\).
### Interpretation in context:
If you interpret this in terms of a runner's speed (as mentioned with "minutes per mile"), a slope of \(10\) means that for every mile (on the x-axis), the runner takes \(10\) minutes (on the y-axis to complete that mile).
Thus, the slope can be described as:
\[
\text{slope} = 10 \text{ minutes per mile}
\]
If you have any other details or questions regarding this scenario, feel free to ask!
\[
\text{slope} (m) = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}
\]
You mentioned two points: \((1.0, 10)\) and \((2.0, 20)\). Let's assign these points:
- Point 1: \((x_1, y_1) = (1.0, 10)\)
- Point 2: \((x_2, y_2) = (2.0, 20)\)
Now substituting into the slope formula:
\[
m = \frac{20 - 10}{2.0 - 1.0} = \frac{10}{1.0} = 10
\]
So, the slope of the line is \(10\).
### Interpretation in context:
If you interpret this in terms of a runner's speed (as mentioned with "minutes per mile"), a slope of \(10\) means that for every mile (on the x-axis), the runner takes \(10\) minutes (on the y-axis to complete that mile).
Thus, the slope can be described as:
\[
\text{slope} = 10 \text{ minutes per mile}
\]
If you have any other details or questions regarding this scenario, feel free to ask!
Answered by
GPT-4o mini
To determine which linear function represents a slope, we need to calculate the slope for the provided tables and sets of points, and then compare them.
### Table 1
| \(x\) | \(y\) |
|--------|---------|
| 3 | -11 |
| 6 | 1 |
| 9 | 13 |
| 12 | 25 |
Calculating the slope using points \((3, -11)\) and \((6, 1)\):
\[
m = \frac{1 - (-11)}{6 - 3} = \frac{1 + 11}{6 - 3} = \frac{12}{3} = 4
\]
### Coordinate Plane 1
The points given are:
- \((0, 3)\)
- \((4, 4)\)
- \((8, 7)\)
Calculating the slope using points \((0, 3)\) and \((4, 4)\):
\[
m = \frac{4 - 3}{4 - 0} = \frac{1}{4}
\]
### Table 2
| \(x\) | \(y\) |
|--------|---------|
| -5 | 32 |
| -1 | 24 |
| 3 | 16 |
| 7 | 8 |
Calculating the slope using points \((-5, 32)\) and \((-1, 24)\):
\[
m = \frac{24 - 32}{-1 - (-5)} = \frac{-8}{4} = -2
\]
### Coordinate Plane 2
The points given are:
- \((2, 0)\)
- \((3, 4)\)
- \((4, 8)\)
Calculating the slope using points \((2, 0)\) and \((3, 4)\):
\[
m = \frac{4 - 0}{3 - 2} = \frac{4}{1} = 4
\]
### Summary of Slopes
1. **Table 1**: Slope = \(4\)
2. **Coordinate Plane 1**: Slope = \(\frac{1}{4}\)
3. **Table 2**: Slope = \(-2\)
4. **Coordinate Plane 2**: Slope = \(4\)
### Conclusion
The linear functions with a **positive slope** are:
- **Table 1**: Slope = \(4\)
- **Coordinate Plane 1**: Slope = \(\frac{1}{4}\)
- **Coordinate Plane 2**: Slope = \(4\)
The only function representing a positive slope is **Table 1** and **Coordinate Plane 2** (both yielding a slope of \(4\)).
**Coordinate Plane 1** also has a positive slope of \(\frac{1}{4}\), while **Table 2** has a negative slope.
If you have further details to clarify or additional questions, feel free to ask!
### Table 1
| \(x\) | \(y\) |
|--------|---------|
| 3 | -11 |
| 6 | 1 |
| 9 | 13 |
| 12 | 25 |
Calculating the slope using points \((3, -11)\) and \((6, 1)\):
\[
m = \frac{1 - (-11)}{6 - 3} = \frac{1 + 11}{6 - 3} = \frac{12}{3} = 4
\]
### Coordinate Plane 1
The points given are:
- \((0, 3)\)
- \((4, 4)\)
- \((8, 7)\)
Calculating the slope using points \((0, 3)\) and \((4, 4)\):
\[
m = \frac{4 - 3}{4 - 0} = \frac{1}{4}
\]
### Table 2
| \(x\) | \(y\) |
|--------|---------|
| -5 | 32 |
| -1 | 24 |
| 3 | 16 |
| 7 | 8 |
Calculating the slope using points \((-5, 32)\) and \((-1, 24)\):
\[
m = \frac{24 - 32}{-1 - (-5)} = \frac{-8}{4} = -2
\]
### Coordinate Plane 2
The points given are:
- \((2, 0)\)
- \((3, 4)\)
- \((4, 8)\)
Calculating the slope using points \((2, 0)\) and \((3, 4)\):
\[
m = \frac{4 - 0}{3 - 2} = \frac{4}{1} = 4
\]
### Summary of Slopes
1. **Table 1**: Slope = \(4\)
2. **Coordinate Plane 1**: Slope = \(\frac{1}{4}\)
3. **Table 2**: Slope = \(-2\)
4. **Coordinate Plane 2**: Slope = \(4\)
### Conclusion
The linear functions with a **positive slope** are:
- **Table 1**: Slope = \(4\)
- **Coordinate Plane 1**: Slope = \(\frac{1}{4}\)
- **Coordinate Plane 2**: Slope = \(4\)
The only function representing a positive slope is **Table 1** and **Coordinate Plane 2** (both yielding a slope of \(4\)).
**Coordinate Plane 1** also has a positive slope of \(\frac{1}{4}\), while **Table 2** has a negative slope.
If you have further details to clarify or additional questions, feel free to ask!
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