How many lines can exist that are perpendicular to a given line and pass through a specific point?

To determine the number of lines that can exist, which are perpendicular to a given line and pass through a specific point, we need to apply a few geometric principles.

1. Perpendicular lines: If two lines are perpendicular, the angle between them is 90 degrees, and the slope of one line is the negative reciprocal of the other.

2. Slope-intercept form: The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

Now, let's break down the steps to find the answer:

Step 1: Find the slope of the given line.
- If the given line is represented by the equation y = mx + b, then the slope (m) can be directly identified.

Step 2: Find the negative reciprocal of the slope from Step 1.
- To find the negative reciprocal, flip the fraction and change its sign. For example, if the slope is 2/3, the negative reciprocal is -3/2.

Step 3: Use the point-slope form to determine the equation of the line passing through the specific point.
- The point-slope form of a line is y - y₁ = m(x - x₁), where (x₁, y₁) represents the coordinates of the specific point, and m is the slope obtained from Step 2.

Step 4: Simplify the equation obtained in Step 3 to slope-intercept form.
- Rearrange the equation to the form y = mx + b, where m is the slope obtained from Step 2, and b is the y-intercept (a constant).

By following these steps, you can determine the equation of the line perpendicular to the given line and passing through the specific point. Since changing the y-intercept can yield different lines, there are infinitely many lines that satisfy these conditions.