To find the equation in slope-intercept form \( y = mx + b \) from the given table, we first need to determine the slope \( m \) and the y-intercept \( b \).
Here are the data points from the table:
- (9, 9)
- (18, -19)
- (27, -26)
Step 1: Calculate the slope \( m \)
The slope formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the first two points (9, 9) and (18, -19):
- \( x_1 = 9 \)
- \( y_1 = 9 \)
- \( x_2 = 18 \)
- \( y_2 = -19 \)
Now, we can substitute in the values: \[ m = \frac{-19 - 9}{18 - 9} = \frac{-28}{9} = -\frac{28}{9} \]
Next, let's use the second and third points (18, -19) and (27, -26):
- \( x_1 = 18 \)
- \( y_1 = -19 \)
- \( x_2 = 27 \)
- \( y_2 = -26 \)
Substituting those values: \[ m = \frac{-26 - (-19)}{27 - 18} = \frac{-26 + 19}{9} = \frac{-7}{9} \]
Now we need to ensure we have consistent slopes. Using the first point and any of the derived points will give us the same slope due to being linear.
Step 2: Find the y-intercept \( b \)
We can use one of the points and the slope to find \( b \). Let's use the point (9, 9).
Substitute into the equation \( y = mx + b \): \[ 9 = -\frac{28}{9}(9) + b \]
Calculating it: \[ 9 = -28 + b \] \[ b = 9 + 28 = 37 \]
Final Equation
Now we have \( m = -\frac{28}{9} \) and \( b = 37 \).
Thus, the equation in slope-intercept form is: \[ y = -\frac{28}{9}x + 37 \]
If preferred, we can also express it as: \[ y = -\frac{28}{9}x + 37 \]