To find the equations for the main hallway and the perpendicular hallway, we can follow these steps:
Step 1: Determine the slope of the main hallway
To find the slope (\(m\)) of the line that passes through points \((-5, 9)\) and \((4, 6)\), we use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \((x_1, y_1) = (-5, 9)\) and \((x_2, y_2) = (4, 6)\).
Substituting these values into the formula gives:
\[ m = \frac{6 - 9}{4 - (-5)} = \frac{-3}{4 + 5} = \frac{-3}{9} = -\frac{1}{3} \]
Step 2: Write the equation of the main hallway
Now that we have the slope, 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 either point. Let's use the point \((-5, 9)\):
\[ y - 9 = -\frac{1}{3}(x + 5) \]
Now we can simplify and convert it to slope-intercept form (y = mx + b):
\[ y - 9 = -\frac{1}{3}x - \frac{5}{3} \]
Adding 9 to both sides:
\[ y = -\frac{1}{3}x - \frac{5}{3} + 9 \]
To combine the constants, convert \(9\) to thirds:
\[ y = -\frac{1}{3}x - \frac{5}{3} + \frac{27}{3} = -\frac{1}{3}x + \frac{22}{3} \]
Thus, the equation of the main hallway is:
\[ y = -\frac{1}{3}x + \frac{22}{3} \]
Step 3: Find the slope of the perpendicular hallway
The slope of the perpendicular hallway is the negative reciprocal of the slope of the main hallway. The slope of the main hallway is \(-\frac{1}{3}\), so the slope of the perpendicular hallway (\(m_{\text{perp}}\)) is:
\[ m_{\text{perp}} = 3 \]
Step 4: Write the equation of the perpendicular hallway
Using the point-slope form of the line again, with the point \((-4, 3)\):
\[ y - 3 = 3(x + 4) \]
Expanding this gives:
\[ y - 3 = 3x + 12 \]
Adding 3 to both sides, we have:
\[ y = 3x + 15 \]
Summary
The equations for the two hallways are:
- Main hallway: \(y = -\frac{1}{3}x + \frac{22}{3}\)
- Perpendicular hallway: \(y = 3x + 15\)