To find the equation of the line $m$ that is orthogonal (perpendicular) to line $l$, we can proceed with the following steps:

Step 1: Identify the Slope of Line $l$

The equation for line $l$ is: $y = -\frac{1}{3}x + 5$ The slope of line $l$, $m_l$, is $-\frac{1}{3}$.

Step 2: Determine the Slope of Line $m$

Since line $m$ is orthogonal to line $l$, its slope $m_m$ must be the negative reciprocal of $m_l$. Therefore, $m_m = -\frac{1}{-1/3} = 3$

Step 3: Use the Point-Slope Form of the Line Equation

We know that line $m$ passes through the point $P(2, 10)$, so we can use the point-slope form of the line equation: $y - y_1 = m(x - x_1)$ where $(x_1, y_1) = (2, 10)$ and $m = 3$.

Substituting these values, we get: $y - 10 = 3(x - 2)$

Step 4: Simplify the Equation

Expanding and simplifying: $y - 10 = 3x - 6$ $y = 3x + 4$

Conclusion

The equation of line $m$ is: $y = 3x + 4$

Would you like further details on any step, or do you have additional questions?

Here are five questions to explore further:

1. What would happen if line $l$ had a different slope?
2. How do we determine the equation of a line given two points?
3. Why is the negative reciprocal used for perpendicular slopes?
4. What if the point $P$ was different—how would that affect the equation of $m$?
5. How can we verify if two lines are perpendicular using their slopes?

Tip: When working with perpendicular lines, remember that the slopes are always negative reciprocals of each other.