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Which tables represent a linear function with the same slope as y = 2 - 3x ?

give 2 examples

7 answers

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 \( y = 2 - 3x \).

try again

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 \(-3\).

a different one for example 2

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!

Omg no negative numbers

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 |

  1. Table 2 (example without negative numbers):

| \( x \) | \( y \) | |---|---| | 0 | 3 | | 1 | 0 |

Again, I apologize for the earlier oversight! Thank you for your patience.