Problem 39 - College Algebra

Question

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither. \(x=4\) \(y=-3\)

Step-by-Step Solution

Answer

The lines are perpendicular.

Step 1: Understanding the Equations

The equations given are \(x = 4\) and \(y = -3\). These can be thought of as special cases of vertical and horizontal lines. The line \(x = 4\) is vertical, crossing the x-axis at 4, and the line \(y = -3\) is horizontal, crossing the y-axis at -3.

Step 2: Sketching the Graph

Draw a graph with x and y axes. The vertical line \(x = 4\) runs parallel to the y-axis at x = 4, and the horizontal line \(y = -3\) runs parallel to the x-axis at y = -3. They intersect at the point (4, -3).

Step 3: Analyzing the Relationship Between the Lines

A vertical line is perpendicular to a horizontal line since they intersect at a right angle (90 degrees). In this case, the vertical line \(x = 4\) and the horizontal line \(y = -3\) are perpendicular.

Key Concepts

Vertical Lines

In the world of graphing linear equations, vertical lines have a distinct characteristic: they run straight up and down, parallel to the y-axis. Such lines are defined by the equation \(x = a\), where \(a\) is a constant. This means every point on a vertical line has the same x-coordinate, but the y-coordinate can vary.

• For example, in the equation \(x = 4\), the line crosses the x-axis at 4, and all points on this line, like (4, 0), (4, 2), or (4, -5), will have 4 as the x-value.
• Vertical lines do not have a slope, or we can say their slope is undefined. This is because in the slope formula \(m = \frac{\text{rise}}{\text{run}}\), the run (change in x) is zero, creating a division by zero.

Though mysterious to some due to their undefined slope, recognizing and understanding the behavior of vertical lines is essential in mastering linear graphs.

Horizontal Lines

Horizontal lines are another special set of linear equations. They stretch from left to right and parallel the x-axis. These are described by the equation \(y = b\), where \(b\) is also a constant, keeping y fixed while the x-value varies.

• The equation \(y = -3\) is representative of a horizontal line crossing the y-axis at -3. Every point along this line, like (0, -3), (5, -3), or (-2, -3), retains -3 as the y-coordinate.
• The slope of a horizontal line is zero. In terms of the slope equation \(m = \frac{\text{rise}}{\text{run}}\), there is no rise (zero change in y), resulting in a slope of zero.

Recognizing these lines in graphs can help in analyzing relationships between different lines and in understanding the geometry of coordinate planes.

Perpendicular Lines

When graphing lines, two lines that intersect at a 90-degree angle are called perpendicular. This geometric relationship is crucial in many mathematical applications. One classic example is when one line is vertical and the other is horizontal.

• In the case of lines \(x = 4\) (vertical line) and \(y = -3\) (horizontal line), their intersection forms a right angle at point (4, -3), demonstrating perpendicularity.
• Mathematically, two lines are perpendicular if the product of their slopes is -1. However, a special scenario exists when dealing with vertical (undefined slope) and horizontal (zero slope) lines, which are always perpendicular due to their geometric alignment.

Understanding perpendicularity is not just about recognizing right angles, but also about appreciating the relationship between the directions of different lines on a graph.