To determine which of the given equations represent the line that passes through the points (9, 3) and (11, 17), we first need to calculate the slope of the line that connects these two points.
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (9, 3) and (11, 17):
\[ m = \frac{17 - 3}{11 - 9} = \frac{14}{2} = 7 \]
Now that we have the slope \( m = 7 \), we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
We can use point (9, 3) in this equation:
\[ y - 3 = 7(x - 9) \]
This simplifies to:
\[ y - 3 = 7x - 63 \implies y = 7x - 60 \]
Now we can examine the given options:
-
\( y - 11 = -7(x - 17) \)
- This one can be simplified using point-slope form, where the slope is negative.
-
\( y - 3 = 7(x - 9) \)
- This matches exactly with what we derived.
-
\( y + 10 = 7(x + 10) \)
- This does not represent the same form and would need further calculation for verification.
Now let's check if option 1 or option 3 can represent a line through our two points.
For option 1: \[ y - 11 = -7(x - 17) \ y - 11 = -7x + 119 \ y = -7x + 130 \]
To check for point (9, 3): \[ 3 = -7(9) + 130 \ 3 = -63 + 130 = 67 \quad (\text{not correct}) \]
For option 3: \[ y + 10 = 7(x + 10) \ y + 10 = 7x + 70 \ y = 7x + 60 \] This is also not correct because the y-intercept does not match.
Given the calculations, the only valid equation derived from original points (9, 3) and (11, 17) using the point-slope format is indeed:
2. \( y - 3 = 7(x - 9) \)
The other two options do not represent the line that goes through the points (9, 3) and (11, 17).