Problem 39 - College Algebra with Corequisite Support

Question

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither. $$ \begin{array}{l} x=4 \\ y=-3 \end{array} $$

Step-by-Step Solution

Answer

The lines are perpendicular to each other.

Step 1: Identify the Lines

The first equation is given as \( x = 4 \). This equation represents a vertical line that passes through all points where the x-coordinate is 4.The second equation is \( y = -3 \). This equation represents a horizontal line that passes through all points where the y-coordinate is -3.

Step 2: Graph the Vertical Line

Plot the vertical line \( x = 4 \) on a coordinate plane. This line will be a straight line parallel to the y-axis, intersecting the x-axis at \( x = 4 \).

Step 3: Graph the Horizontal Line

Plot the horizontal line \( y = -3 \) on the same coordinate plane. This line will be a straight line parallel to the x-axis, intersecting the y-axis at \( y = -3 \).

Step 4: Analyze the Relationship Between the Lines

A vertical line \( x = 4 \) and a horizontal line \( y = -3 \) meet at a right angle. Therefore, these lines are perpendicular to each other. Vertical and horizontal lines always intersect perpendicularly.

Key Concepts

Graphing Linear Equations

Graphing linear equations allows us to visualize relationships between variables by plotting their solutions. In coordinate geometry, a linear equation can be plotted as a straight line on the coordinate plane. This line represents all possible solutions of the equation, showing how one variable affects the other.
For linear equations in two variables, such as the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, the graph is a line spanning across the plane. Here, every point on the line is a solution to the equation.

• **Slope (m)**: Indicates the steepness and direction of the line. A positive slope rises, while a negative slope falls.
• **Y-intercept (b)**: The point where the line crosses the y-axis. It's the value of y when x is zero.

For equations involving only one variable, like \( x = 4 \) or \( y = -3 \), these form vertical and horizontal lines, respectively, which are special cases in graphing.

Perpendicular Lines

Perpendicular lines are lines that intersect each other at a right angle, forming 90 degrees. Understanding the concept of perpendicularity is crucial in geometry, as it defines many properties of shapes.
Two lines in the plane are perpendicular when the product of their slopes is -1. For example, a line with a slope of \( m \) and another line with a slope of \( -1/m \) will be perpendicular.
In the context of vertical and horizontal lines, their slopes are undefined and zero, respectively. A vertical line, \( x = a \), runs parallel to the y-axis, while a horizontal line, \( y = b \), runs parallel to the x-axis. When these lines intersect, they do so at right angles, making them always perpendicular as seen in our exercise example with \( x = 4 \) and \( y = -3 \).

Vertical and Horizontal Lines

Vertical and horizontal lines are fundamental elements in coordinate geometry, providing the basis for understanding two-dimensional space on the plane.

• **Vertical Lines**: A vertical line is described by an equation of the form \( x = a \), where every point on the line has the same x-coordinate. This line is parallel to the y-axis and can be thought of as infinite in length up and down.
• **Horizontal Lines**: A horizontal line is represented by an equation like \( y = b \), where all points share the same y-coordinate. This line stretches left to right, indefinitely, parallel to the x-axis.

In any graph, vertical and horizontal lines are straightforward to interpret and plot, making them a go-to for simplifying complex functions or showcasing intercepts and specific values clearly. These lines' unique property of always intersecting perpendicularly is central to many geometric proofs and practical applications.