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 \(y=8\); containing the point (3,4)

Short answer

The equation is \(x = 3\).

Step-by-step solution

Identify the Slope of the Given Line

The given line is in the slope-intercept form, which is generally written as \(y = mx + b\), where \(m\) is the slope. For the line \(y = 8\), the slope \(m = 0\) because the line is horizontal.

Find the Slope of the Perpendicular Line

Perpendicular lines have slopes that are negative reciprocals of each other. The slope of the given line is 0. Therefore, the slope of the line perpendicular to it will be undefined, which means it is a vertical line.

Use the Point to Find the Equation of the Line

A vertical line passing through a point (3, 4) has an equation where \(x\) is constant. Since it passes through the point (3, 4), the equation of the line is \(x = 3\).

Key concepts

Slope-Intercept Form

When we talk about the equation of a line, one of the most common forms you'll encounter is the slope-intercept form. This form is written as:

y = mx + b
In this formula, m represents the slope of the line, which shows how steep the line is. The b term is the y-intercept, where the line crosses the y-axis.

To identify the slope and y-intercept from an equation, look for the coefficient of x (that's your slope) and the constant term (that's your y-intercept). For instance, in the equation y = 2x + 3, the slope m is 2, and the y-intercept b is 3.

Understanding the slope-intercept form is crucial because it builds a foundation for grasping more complex line equations, including those involving perpendicular or vertical lines.

Perpendicular Lines

Perpendicular lines are lines that intersect to form a right angle, which is 90 degrees. One interesting aspect of perpendicular lines is their slopes. The slopes of two perpendicular lines are negative reciprocals of each other.

For example, if one line has a slope of m1, the perpendicular line's slope m2 would be computed as:
m2 = -1/m1

If the slope of a line is horizontal (like m = 0 ), a line perpendicular to it will be vertical, leading to an undefined slope. That's precisely what makes vertical lines special in this scenario. Knowing this helps, for example, when deriving perpendicular lines from horizontal ones like we did in the given exercise.

Vertical Line

A vertical line has the unique property of having an undefined slope. This is because for vertical lines, the change in the y-coordinate can occur without any change in the x-coordinate.

Mathematically, this translates to dividing by zero in the slope formula, making it undefined. A vertical line's equation is straightforward: it's written in the form x = a constant value.

For instance, if we need a vertical line passing through the point (3, 4), we simply write:
x = 3

This form of line equation indicates that no matter what y is, x will always be 3. Vertical lines are essential in geometry and algebra because they set a foundation for understanding more complex concepts involving slopes and line equations.