What is the equation of a line that is perpendicular to the line defined by y = 4x + 5 and passes through the point (8, 3)?

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To find the equation of a line that is perpendicular to the line given by the equation y = 4x + 5 and passes through the point (8, 3), we can follow these steps:

  1. Determine the slope of the given line: The equation of the line is in the slope-intercept form y = mx + b, where m represents the slope. From the equation y = 4x + 5, we see that the slope (m) is 4.
  2. Find the slope of the perpendicular line: The slopes of two perpendicular lines are negative reciprocals of each other. Therefore, to find the slope of the line we want, we take the negative reciprocal of 4, which is -1/4.
  3. Use the point-slope form of the equation: With the slope of the line we are looking for being -1/4 and it passing through the point (8, 3), we can use the point-slope form of the equation, which is given by:
    4. y - y_1 = m(x - x_1)

    Here (x_1, y_1) is the point the line passes through, so:

    y - 3 = -1/4(x - 8)
  4. Simplify the equation: Now we can simplify the equation to put it in slope-intercept form. Distributing the slope:
    5. y - 3 = -1/4x + 2

    Next, add 3 to both sides:

    y = -1/4x + 2 + 3
    y = -1/4x + 5

Final Answer: The equation of the line that is perpendicular to y = 4x + 5 and passes through the point (8, 3) is:

y = -1/4x + 5

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