Let’s complete the work step by step based on your problem where a line passes through the point (12, –5) with a slope of \( \frac{1}{3} \).
1. Substitute known values for \( m \), \( x_1 \), and \( y_1 \):
Given:
- Slope \( m = \frac{1}{3} \)
- Point \( (x_1, y_1) = (12, -5) \)
Using the point-slope form of the line equation, which is: \[ y - y_1 = m(x - x_1) \]
Substituting the known values: \[ y - (-5) = \frac{1}{3}(x - 12) \] Which simplifies to: \[ y + 5 = \frac{1}{3}(x - 12) \]
2. Distribute the slope through the parentheses:
Now we distribute \( \frac{1}{3} \) on the right side: \[ y + 5 = \frac{1}{3}x - \frac{1}{3} \cdot 12 \]
Calculating \( \frac{1}{3} \cdot 12 \): \[ \frac{1}{3} \cdot 12 = 4 \]
Thus, the equation becomes: \[ y + 5 = \frac{1}{3}x - 4 \]
3. Distribute the slope through the parentheses (again):
This step is not necessary, as we’ve already performed the distribution. So, we can proceed directly to the next step.
4. Solve for the y-variable:
Now, isolate \( y \): \[ y = \frac{1}{3}x - 4 - 5 \]
Simplifying: \[ y = \frac{1}{3}x - 9 \]
Final Answer:
So, the equation of the line passing through the point (12, –5) with a slope of \( \frac{1}{3} \) is: \[ y = \frac{1}{3}x - 9 \]