To find the linear equation that relates \( x \) and \( y \), we can first examine the values provided in the table:
x: 2, 3, 4, 5
y: -36, -54, -72, -90
Next, we can find the rate of change (slope) between the \( x \) and \( y \) values. We can calculate the change in \( y \) for each increment in \( x \):
- From \( x = 2 \) to \( x = 3 \): \[ \Delta y = -54 - (-36) = -18 \]
- From \( x = 3 \) to \( x = 4 \): \[ \Delta y = -72 - (-54) = -18 \]
- From \( x = 4 \) to \( x = 5 \): \[ \Delta y = -90 - (-72) = -18 \]
The slope \( m \) is consistent and equal to \(-18\).
Next, we can use the points to find the y-intercept. To do this, we can use the point \( (2, -36) \):
The linear equation can be expressed in the form: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. Substituting the slope and the point we can find \( b \) as follows:
\[ -36 = (-18)(2) + b \] \[ -36 = -36 + b \] \[ b = 0 \]
Now we can write the linear equation: \[ y = -18x \]
So the final answer is: \[ y = -18x \]