To find the linear equation that fits the given values in the table, we can start by examining the changes in x and y.
Given Data:
x | y
-----
3 | 14
4 | 4
5 | -6
6 | -16
Find the Differences:
To determine the slope (m), we can calculate the change in y when x changes.
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For \( x = 3 \) to \( x = 4 \) (change in x = 1):
- \( y \) changes from 14 to 4:
- Change in \( y = 4 - 14 = -10 \)
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For \( x = 4 \) to \( x = 5 \) (change in x = 1):
- \( y \) changes from 4 to -6:
- Change in \( y = -6 - 4 = -10 \)
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For \( x = 5 \) to \( x = 6 \) (change in x = 1):
- \( y \) changes from -6 to -16:
- Change in \( y = -16 - (-6) = -10 \)
The change in \( y \) for each increase in \( x \) by 1 is consistently -10. Therefore, the slope \( m \) is -10.
Find the y-intercept:
We will use one of the points to find the y-intercept \( b \). We can use the point \( (3, 14) \) and the slope -10.
The general form of the linear equation is: \[ y = mx + b \]
Substituting \( m = -10 \) and \( (x, y) = (3, 14) \): \[ 14 = -10(3) + b \] \[ 14 = -30 + b \] \[ b = 14 + 30 \] \[ b = 44 \]
Final Equation:
Thus, the linear equation that describes the rule for the table is: \[ y = -10x + 44 \]
So in the form requested: \[ y = -10x + 44 \]
Therefore, the completed statement is:
- \( y = -10x + 44 \)