To find the linear equation that relates \(x\) and \(y\), we first look for a pattern in the values.
Let's analyze the change in \(y\) as \(x\) increases:
- When \(x = 4\), \(y = -32\)
- When \(x = 5\), \(y = -40\) (change in \(y = -40 - (-32) = -8\))
- When \(x = 6\), \(y = -48\) (change in \(y = -48 - (-40) = -8\))
- When \(x = 7\), \(y = -56\) (change in \(y = -56 - (-48) = -8\))
The change in \(y\) is consistent at \(-8\) for each increase of \(1\) in \(x\). This suggests a linear relationship with a slope of \(-8\).
Now we can express this linear relationship in the form of the equation: \[ y = mx + b \] where \(m\) is the slope and \(b\) is the y-intercept.
Using the slope \(m = -8\), we can write: \[ y = -8x + b \] To find \(b\), we can use one of the points in the table. Let's use the point \( (4, -32) \).
Substituting \(x = 4\) and \(y = -32\): \[ -32 = -8(4) + b \] \[ -32 = -32 + b \] \[ b = 0 \]
Therefore, the equation is: \[ y = -8x \]
Thus, the completed equation is: \[ y = -8x \]