Final answer:
To find the equation of a line that is perpendicular to another line and passes through a given point, we need to determine the slope of the given line and then find the negative reciprocal of that slope. The equation of the line that is perpendicular to y + 1 = -3(x - 5) and passes through the point (4, -6) is x - 3y = 22.
Step-by-step explanation:
To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line and then find the negative reciprocal of that slope. The given line is y + 1 = -3(x - 5), which is in the form y = mx + b, where m is the slope. So, the slope of the given line is -3. The negative reciprocal of -3 is 1/3.
Since the line we are looking for passes through the point (4, -6), we can use the point-slope form of the equation of a line: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Substituting the values, we get y - (-6) = 1/3(x - 4). Simplifying this equation, we get y + 6 = 1/3x - 4/3. Multiplying both sides by 3 to eliminate the fraction, we get 3(y + 6) = x - 4. Simplifying further, we get 3y + 18 = x - 4. Rearranging the equation, we get x - 3y = 22. Therefore, the equation of the line that is perpendicular to y + 1 = -3(x - 5) and passes through the point (4, -6) is x - 3y = 22.