To find the equation of a line perpendicular to $y = -3x + 5$ and passing through the point $(9, -1)$, follow these steps:

Step 1: Determine the slope of the given line

The slope of the given line $y = -3x + 5$ is $m = -3$.

Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to another line is the negative reciprocal of the original slope. For $m = -3$, the perpendicular slope $m_\perp$ is: $m_\perp = \frac{-1}{-3} = \frac{1}{3}.$

Step 3: Use the point-slope form of the line equation

The point-slope form of a line is: $y - y_1 = m(x - x_1),$ where $(x_1, y_1)$ is a point on the line, and $m$ is the slope.

Substitute $m = \frac{1}{3}$ and $(x_1, y_1) = (9, -1)$: $y - (-1) = \frac{1}{3}(x - 9).$

Simplify: $y + 1 = \frac{1}{3}(x - 9).$

Step 4: Rearrange to slope-intercept form (optional)

Distribute $\frac{1}{3}$ on the right-hand side: $y + 1 = \frac{1}{3}x - 3.$

Subtract 1 from both sides: $y = \frac{1}{3}x - 4.$

Final Equation

The equation of the line perpendicular to $y = -3x + 5$ and passing through $(9, -1)$ is: $y = \frac{1}{3}x - 4.$

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Related Questions:

1. What is the slope of a line parallel to $y = -3x + 5$?
2. How do you find the y-intercept of a line given its slope and a point?
3. Can you explain how to derive the point-slope form of a line?
4. How do you verify if two lines are perpendicular?
5. What are the steps to convert from point-slope to standard form?

Tip:

Always remember that perpendicular lines have slopes that multiply to $-1$.