Question
Which tables represent a linear function with the same slope as y = 2 - 3x ?
give 2 examples
give 2 examples
Answers
Answer
try again
Answer
a different one for example 2
Answer
Omg no negative numbers
Answered by
GPT-4o mini
To determine which tables represent a linear function with the same slope as the function \( y = 2 - 3x \), we first need to identify the slope of the given linear function.
The given equation can be rewritten in slope-intercept form \( y = mx + b \) as:
\[
y = -3x + 2
\]
Here, the slope \( m \) is \(-3\).
Now we need to find two examples of tables that represent a linear function with a slope of \(-3\). For each entry in the table, we can use the relationship \( y - y_1 = m(x - x_1) \) where \( m = -3 \).
### Example 1:
| \( x \) | \( y \) |
|---|---|
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
| 3 | -7 |
**Calculations**:
- From \( x = 0 \) to \( x = 1 \): \( y = 2 - 3(1) = -1 \) (slope calculated as \(\frac{-1 - 2}{1 - 0} = -3\))
- From \( x = 1 \) to \( x = 2 \): \( y = -1 - 3(1) = -4 \) (slope calculated as \(\frac{-4 - (-1)}{2 - 1} = -3\))
- From \( x = 2 \) to \( x = 3 \): \( y = -4 - 3(1) = -7 \) (slope calculated as \(\frac{-7 - (-4)}{3 - 2} = -3\))
### Example 2:
| \( x \) | \( y \) |
|---|---|
| -1 | 5 |
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
**Calculations**:
- From \( x = -1 \) to \( x = 0 \): \( y = 5 - 3(1) = 2 \) (slope calculated as \(\frac{2 - 5}{0 - (-1)} = -3\))
- From \( x = 0 \) to \( x = 1 \): \( y = 2 - 3(1) = -1 \) (slope calculated as \(\frac{-1 - 2}{1 - 0} = -3\))
- From \( x = 1 \) to \( x = 2 \): \( y = -1 - 3(1) = -4 \) (slope calculated as \(\frac{-4 - (-1)}{2 - 1} = -3\))
Both of these tables represent linear functions with a slope of \(-3\), the same as the slope of the function \( y = 2 - 3x \).
The given equation can be rewritten in slope-intercept form \( y = mx + b \) as:
\[
y = -3x + 2
\]
Here, the slope \( m \) is \(-3\).
Now we need to find two examples of tables that represent a linear function with a slope of \(-3\). For each entry in the table, we can use the relationship \( y - y_1 = m(x - x_1) \) where \( m = -3 \).
### Example 1:
| \( x \) | \( y \) |
|---|---|
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
| 3 | -7 |
**Calculations**:
- From \( x = 0 \) to \( x = 1 \): \( y = 2 - 3(1) = -1 \) (slope calculated as \(\frac{-1 - 2}{1 - 0} = -3\))
- From \( x = 1 \) to \( x = 2 \): \( y = -1 - 3(1) = -4 \) (slope calculated as \(\frac{-4 - (-1)}{2 - 1} = -3\))
- From \( x = 2 \) to \( x = 3 \): \( y = -4 - 3(1) = -7 \) (slope calculated as \(\frac{-7 - (-4)}{3 - 2} = -3\))
### Example 2:
| \( x \) | \( y \) |
|---|---|
| -1 | 5 |
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
**Calculations**:
- From \( x = -1 \) to \( x = 0 \): \( y = 5 - 3(1) = 2 \) (slope calculated as \(\frac{2 - 5}{0 - (-1)} = -3\))
- From \( x = 0 \) to \( x = 1 \): \( y = 2 - 3(1) = -1 \) (slope calculated as \(\frac{-1 - 2}{1 - 0} = -3\))
- From \( x = 1 \) to \( x = 2 \): \( y = -1 - 3(1) = -4 \) (slope calculated as \(\frac{-4 - (-1)}{2 - 1} = -3\))
Both of these tables represent linear functions with a slope of \(-3\), the same as the slope of the function \( y = 2 - 3x \).
Answered by
GPT-4o mini
To find tables that represent linear functions with the same slope as the equation \( y = 2 - 3x \) (which has a slope of \(-3\)), we need to generate values that maintain the constant rate of change of \(-3\).
### Example 1:
**Table 1:**
| \( x \) | \( y \) |
|---|---|
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
| 3 | -7 |
**Calculations:**
- For \( x = 0 \): \( y = 2 \)
- For \( x = 1 \): \( y = 2 - 3(1) = -1 \)
- For \( x = 2 \): \( y = 2 - 3(2) = -4 \)
- For \( x = 3 \): \( y = 2 - 3(3) = -7 \)
The difference in \( y \) values as \( x \) increments by 1 is always \(-3\), confirming the same slope.
### Example 2:
**Table 2:**
| \( x \) | \( y \) |
|---|---|
| -1 | 5 |
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
**Calculations:**
- For \( x = -1 \): \( y = 2 - 3(-1) = 5 \)
- For \( x = 0 \): \( y = 2 - 3(0) = 2 \)
- For \( x = 1 \): \( y = 2 - 3(1) = -1 \)
- For \( x = 2 \): \( y = 2 - 3(2) = -4 \)
Again, the difference in \( y \) values as \( x \) increments by 1 is \(-3\).
These examples illustrate that both tables represent linear functions with a slope of \(-3\).
### Example 1:
**Table 1:**
| \( x \) | \( y \) |
|---|---|
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
| 3 | -7 |
**Calculations:**
- For \( x = 0 \): \( y = 2 \)
- For \( x = 1 \): \( y = 2 - 3(1) = -1 \)
- For \( x = 2 \): \( y = 2 - 3(2) = -4 \)
- For \( x = 3 \): \( y = 2 - 3(3) = -7 \)
The difference in \( y \) values as \( x \) increments by 1 is always \(-3\), confirming the same slope.
### Example 2:
**Table 2:**
| \( x \) | \( y \) |
|---|---|
| -1 | 5 |
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
**Calculations:**
- For \( x = -1 \): \( y = 2 - 3(-1) = 5 \)
- For \( x = 0 \): \( y = 2 - 3(0) = 2 \)
- For \( x = 1 \): \( y = 2 - 3(1) = -1 \)
- For \( x = 2 \): \( y = 2 - 3(2) = -4 \)
Again, the difference in \( y \) values as \( x \) increments by 1 is \(-3\).
These examples illustrate that both tables represent linear functions with a slope of \(-3\).
Answered by
GPT-4o mini
Sure! Let's create a different linear function table for Example 2 while ensuring it still has a slope of \(-3\).
### Example 1 (repeated):
**Table 1:**
| \( x \) | \( y \) |
|---|---|
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
| 3 | -7 |
### Example 2 (new):
**Table 2:**
| \( x \) | \( y \) |
|---|---|
| -2 | 8 |
| -1 | 5 |
| 0 | 2 |
| 1 | -1 |
**Calculations:**
- For \( x = -2 \): \( y = 2 - 3(-2) = 8 \)
- For \( x = -1 \): \( y = 2 - 3(-1) = 5 \)
- For \( x = 0 \): \( y = 2 - 3(0) = 2 \)
- For \( x = 1 \): \( y = 2 - 3(1) = -1 \)
### Check the slope:
- From \( (-2, 8) \) to \( (-1, 5) \): \(\frac{5 - 8}{-1 - (-2)} = \frac{-3}{1} = -3\)
- From \( (-1, 5) \) to \( (0, 2) \): \(\frac{2 - 5}{0 - (-1)} = \frac{-3}{1} = -3\)
- From \( (0, 2) \) to \( (1, -1) \): \(\frac{-1 - 2}{1 - 0} = \frac{-3}{1} = -3\)
Both tables now represent linear functions with the same slope of \(-3\). Thank you for your patience!
### Example 1 (repeated):
**Table 1:**
| \( x \) | \( y \) |
|---|---|
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
| 3 | -7 |
### Example 2 (new):
**Table 2:**
| \( x \) | \( y \) |
|---|---|
| -2 | 8 |
| -1 | 5 |
| 0 | 2 |
| 1 | -1 |
**Calculations:**
- For \( x = -2 \): \( y = 2 - 3(-2) = 8 \)
- For \( x = -1 \): \( y = 2 - 3(-1) = 5 \)
- For \( x = 0 \): \( y = 2 - 3(0) = 2 \)
- For \( x = 1 \): \( y = 2 - 3(1) = -1 \)
### Check the slope:
- From \( (-2, 8) \) to \( (-1, 5) \): \(\frac{5 - 8}{-1 - (-2)} = \frac{-3}{1} = -3\)
- From \( (-1, 5) \) to \( (0, 2) \): \(\frac{2 - 5}{0 - (-1)} = \frac{-3}{1} = -3\)
- From \( (0, 2) \) to \( (1, -1) \): \(\frac{-1 - 2}{1 - 0} = \frac{-3}{1} = -3\)
Both tables now represent linear functions with the same slope of \(-3\). Thank you for your patience!
Answered by
GPT-4o mini
I apologize for misunderstanding your request! Let's create two tables with non-negative \( y \) values that still represent functions with a slope of \(-3\).
### Example 1:
**Table 1:**
| \( x \) | \( y \) |
|---|---|
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
| 3 | -7 |
(Note: This table has negative values; I will ensure the next example does not.)
### Example 2 (new and non-negative):
Let's adjust our approach to keep \( y \) values non-negative while having a linear relationship with a slope of \(-3\).
**Table 2:**
| \( x \) | \( y \) |
|---|---|
| 0 | 6 |
| 1 | 3 |
| 2 | 0 |
**Calculations:**
- For \( x = 0 \): \( y = 6 \)
- For \( x = 1 \): \( y = 6 - 3(1) = 3 \)
- For \( x = 2 \): \( y = 6 - 3(2) = 0 \)
### Check the slope:
- From \( (0, 6) \) to \( (1, 3) \): \(\frac{3 - 6}{1 - 0} = \frac{-3}{1} = -3\)
- From \( (1, 3) \) to \( (2, 0) \): \(\frac{0 - 3}{2 - 1} = \frac{-3}{1} = -3\)
Now, both tables have only non-negative \( y \) values:
1. **Table 1 (valid):**
| \( x \) | \( y \) |
|---|---|
| 0 | 6 |
| 1 | 3 |
| 2 | 0 |
2. **Table 2 (example without negative numbers):**
| \( x \) | \( y \) |
|---|---|
| 0 | 3 |
| 1 | 0 |
Again, I apologize for the earlier oversight! Thank you for your patience.
### Example 1:
**Table 1:**
| \( x \) | \( y \) |
|---|---|
| 0 | 2 |
| 1 | -1 |
| 2 | -4 |
| 3 | -7 |
(Note: This table has negative values; I will ensure the next example does not.)
### Example 2 (new and non-negative):
Let's adjust our approach to keep \( y \) values non-negative while having a linear relationship with a slope of \(-3\).
**Table 2:**
| \( x \) | \( y \) |
|---|---|
| 0 | 6 |
| 1 | 3 |
| 2 | 0 |
**Calculations:**
- For \( x = 0 \): \( y = 6 \)
- For \( x = 1 \): \( y = 6 - 3(1) = 3 \)
- For \( x = 2 \): \( y = 6 - 3(2) = 0 \)
### Check the slope:
- From \( (0, 6) \) to \( (1, 3) \): \(\frac{3 - 6}{1 - 0} = \frac{-3}{1} = -3\)
- From \( (1, 3) \) to \( (2, 0) \): \(\frac{0 - 3}{2 - 1} = \frac{-3}{1} = -3\)
Now, both tables have only non-negative \( y \) values:
1. **Table 1 (valid):**
| \( x \) | \( y \) |
|---|---|
| 0 | 6 |
| 1 | 3 |
| 2 | 0 |
2. **Table 2 (example without negative numbers):**
| \( x \) | \( y \) |
|---|---|
| 0 | 3 |
| 1 | 0 |
Again, I apologize for the earlier oversight! Thank you for your patience.
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