To find the linear equation that fits the data in the table, we first observe the relationship between \( x \) and \( y \).
The values for \( x \) and \( y \) are:
- When \( x = 4 \), \( y = 88 \)
- When \( x = 5 \), \( y = 92 \)
- When \( x = 6 \), \( y = 96 \)
- When \( x = 7 \), \( y = 100 \)
Now, we can determine the slope (\( m \)) of the linear equation using the changes in \( y \) and \( x \):
From \( x = 4 \) to \( x = 5 \): \[ \Delta y = 92 - 88 = 4 \] \[ \Delta x = 5 - 4 = 1 \] So, the slope \( m = \frac{\Delta y}{\Delta x} = \frac{4}{1} = 4 \).
Next, we can use one of the points to write the equation in point-slope form, then convert it to slope-intercept form. Using the point \( (4, 88) \):
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Start with the point-slope form: \[ y - 88 = 4(x - 4) \]
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Distribute the slope: \[ y - 88 = 4x - 16 \]
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Add 88 to both sides to isolate \( y \): \[ y = 4x - 16 + 88 \] \[ y = 4x + 72 \]
Thus, the equation that gives the rule for this table is: \[ y = 4x + 72 \]