How can I find the equation of a line that is perpendicular to a given line and passes through a specific point?

To find the equation of a line that is perpendicular to a given line and passes through a specific point, follow these steps:

1. Identify the slope of the given line: The equation of the line should be in the slope-intercept form, which is y = mx + b, where m is the slope. If the line is not in this form, you may need to rearrange it to find the slope.
2. Determine the slope of the perpendicular line: Lines that are perpendicular to each other have slopes that are negative reciprocals. This means that if the slope of the given line is m, the slope of the perpendicular line will be -1/m.
3. Use the point-slope form of the equation of a line: The point-slope form is y - y_1 = m(x - x_1), where (x_1, y_1) is the point through which the line passes, and m is the slope you found in the previous step.
4. Substitute the values: Plug in the coordinates of the given point and the slope of the perpendicular line into the point-slope formula.
5. Simplify and rearrange: If necessary, simplify the equation to get it into slope-intercept form or any other desired format.

Example: Let’s say the given line is y = 2x + 3 and the point is (1, 4).

1. The slope of the given line is m = 2.
2. The slope of the perpendicular line is -1/2.
3. Using the point-slope form: y - 4 = -1/2(x - 1).
4. Distributing gives: y - 4 = -1/2x + 1/2.
5. Finally, adding 4 to both sides: y = -1/2x + 4.5.

Thus, the equation of the line that is perpendicular to the given line and passes through the point (1, 4) is y = -1/2x + 4.5.