Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Perpendicular to the line \(x=8\); containing the point (3,4)

Short answer

The equation is y = 4.

Step-by-step solution

Identify the characteristics of the given line

The given line is vertical with the equation x = 8. Vertical lines have an undefined slope.

Determine the slope of the perpendicular line

A line that is perpendicular to a vertical line is horizontal. Horizontal lines have a slope of 0.

Use the point-slope form

Use the point-slope form of the equation of a line, where the slope (m) is 0 and the point is (3, 4). The point-slope form is: \[ y - y_1 = m(x - x_1) \] Substitute \( m = 0 \), \( x_1 = 3 \), and \( y_1 = 4 \).

Simplify the equation

The equation becomes \[ y - 4 = 0(x - 3) \] which simplifies to \[ y - 4 = 0 \].

Write the final equation

Solving the simplified equation for y gives: \[ y = 4 \].

Key concepts

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle, or 90 degrees. When two lines are perpendicular, the product of their slopes is -1. For example, if one line has a slope of 2, then the line perpendicular to it will have a slope of -1/2. This property is very useful when you need to find an equation of a line perpendicular to another.

In the exercise, you were given a vertical line with the equation x = 8. Notice that vertical lines have an undefined slope. Therefore, any line perpendicular to x = 8 must be horizontal, since horizontal lines have a slope of 0.

Horizontal Line

A horizontal line runs from left to right and is parallel to the x-axis. It has a constant y-value for any x-value. Therefore, the equation of a horizontal line is always in the form of y = c, where c is the y-value through which the line passes.

In the given problem, the line perpendicular to x = 8 must be horizontal. This means our line is parallel to the x-axis and has a constant y-value. Given that the line must pass through the point (3, 4), we can deduce that c = 4. Thus, the equation of the line is y = 4.

Point-Slope Form

The point-slope form of a line’s equation is useful when you have a point and the slope of the line. It can be written as: y - y_1 = m(x - x_1).
Here, (x_1, y_1) is the point through which the line passes, and m is the slope of the line.

In the problem, we used point-slope form to find the equation. We had the point (3, 4) and the slope m = 0, since the line is horizontal. Plugging these values into the point-slope formula, we get: y - 4 = 0(x - 3). This simplifies to y - 4 = 0, which further simplifies to y = 4.

Slope-Intercept Form

The slope-intercept form of a line’s equation is y = mx + b, where m is the slope and b is the y-intercept of the line. This form is particularly useful for quickly identifying the slope and intercept of the line.

For horizontal lines, because the slope m is 0, the equation simplifies to y = b. That is, the y-value is constant for any value of x. In the given exercise, since the line passes through the point (3, 4) and is horizontal, the y-intercept b is 4. Thus, the equation in slope-intercept form is simply y = 4.