Problem 36 - Algebra 2

Question

Graph the line that satisfies each set of conditions. passes through \((3,3),\) perpendicular to graph of \(y=3\)

Step-by-Step Solution

Answer

The line is vertical and has the equation \( x = 3 \).

Step 1: Understand the given line

The line given by the equation \( y = 3 \) is a horizontal line that crosses the \( y \)-axis at 3.

Step 2: Determine the slope of a perpendicular line

A line that is perpendicular to a horizontal line \( y = 3 \) must be vertical, as horizontal and vertical lines are perpendicular to each other. Therefore, its slope is undefined.

Step 3: Use the point to determine the equation of the vertical line

Since the line must be vertical and pass through the point (3,3), the equation of the line is \( x = 3 \). Vertical lines have equations in the form \( x = c \), where \( c \) is a constant.

Key Concepts

Coordinate Geometry Essentials

Coordinate geometry is a branch of mathematics that combines algebra and geometry to describe the properties and relationships of points, lines, and figures in a plane using coordinates. The coordinate system consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). When plotting points, you use an ordered pair

• The first number represents the position along the x-axis.
• The second number represents the position on the y-axis.

In the plane, a point like \((3,3)\) represents a specific location where both x and y values are equal to 3.
Coordinate geometry is also crucial for understanding how lines and shapes interact with one another, whether they intersect, are parallel, or perpendicular.

Understanding Perpendicular Lines

Lines are perpendicular if they intersect at a right angle, which is 90 degrees. On the coordinate plane, horizontal and vertical lines are the most straightforward examples of perpendicular lines. A horizontal line runs parallel to the x-axis, while a vertical line runs parallel to the y-axis.
For non-vertical and non-horizontal lines:

• Perpendicular lines have slopes that are negative reciprocals of each other. For example, if one line has a slope of \( m \), the perpendicular line will have a slope of \(-\frac{1}{m}\).
In this exercise, the line described by \( y = 3 \) is horizontal, and therefore, any line perpendicular to it must be vertical, with an undefined slope.

Equations of Lines

The equation of a line is a mathematical statement that describes all the points along a line in the coordinate plane. The most common forms of line equations are:

• Slope-intercept form: \( y = mx + b \)
• Point-slope form: \( y - y_1 = m(x - x_1) \)

For vertical lines, these forms are not applicable because the slope \( m \) is undefined.
Vertical lines have their own form, \( x = c \), where \( c \) is a constant. This equation shows that in a vertical line, the x-coordinate never changes, regardless of the y-values.

Exploring Vertical Lines

Vertical lines are unique because they run parallel to the y-axis. They do not tilt; instead, they are perfectly upright. This implies that their slope is undefined, which means you cannot express their slope with a conventional number.
Consider a vertical line passing through \((3,3)\). The equation of such a line would be \( x = 3 \). This indicates that at every point along the line, the x-coordinate remains constant at 3, while the y-coordinate can take any value.

• Vertical lines are also known for being perpendicular to every horizontal line.
• They intersect the x-axis at precisely one point if they do not lie along it.

Understanding vertical lines is crucial for problems involving perpendicularity in coordinate geometry.