Problem 52 - Algebra and Trigonometry

Question

Find the slope and y-intercept of the line, and draw its graph. $$ x=-5 $$

Step-by-Step Solution

Answer

Slope is undefined; no y-intercept.

Step 1: Understand the Equation

The given equation is \( x = -5 \). This is a vertical line where the x-coordinate of every point on the line is -5.

Step 2: Analyze the Slope

For a vertical line like \( x = -5 \), the slope is undefined because the line goes straight up and down.

Step 3: Identify the Y-Intercept

A vertical line does not have a y-intercept because it does not cross the y-axis at any point.

Step 4: Drawing the Line on the Graph

To graph the line \( x = -5 \), draw a straight vertical line that intersects the x-axis at \( -5 \).

Key Concepts

Slope of a Line

The slope of a line in general measures how steep a line is. It tells us how much the line rises or falls as we move from left to right across a graph. The formula to find the slope, commonly represented as \( m \), is calculated using two points on the line: \((x_1, y_1)\) and \((x_2, y_2)\). The formula is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

A positive slope means the line rises as it moves from left to right, and a negative slope means it falls. If the line is perfectly horizontal, the slope is 0. However, when it comes to vertical lines, like in the equation \( x = -5 \), the slope is undefined. This is because you would be dividing by zero in the slope calculation, as the difference in \( x \)-values is zero, causing the division to not have a unique value.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It is expressed as a coordinate \((0, b)\), where \( b \) is the y-intercept value. This concept is used to find where the line "hits" the y-axis when extended, and it is particularly important for graphing and understanding the line's placement in coordinate space.

• For most lines, finding the y-intercept is simply a matter of seeing where the line touches the y-axis.
• In the form \( y = mx + b \), "\( b \)" is often added directly by placing the equation in slope-intercept form.

However, in the case of a vertical line such as \( x = -5 \), there is no y-intercept. This occurs because the line is vertical and never passes through the y-axis. The line instead runs parallel to it.

Vertical Line

A vertical line is a straight line that goes up and down in a graph, and it is described by an equation in the form \( x = a \), where \( a \) is the constant x-coordinate of every point on the line. One key characteristic of a vertical line is that it has an undefined slope, as mentioned earlier, due to no horizontal change—it neither rises nor falls as it goes straight up or down.

• Vertical lines have the same x-value at every point, illustrating that all points lie directly above or below each other.
• This can make vertical lines useful for marking specific locations on the x-axis or emphasizing boundaries in a graph.

An example often seen is \( x = -5 \), which visually on the graph appears as a line intersecting the x-axis at -5 and extending infinitely up and down without ever meeting the y-axis.

Graphing

Graphing is the method of illustrating equations or functions visually using a coordinate system. In a typical XY-chart, the x-axis runs horizontally, and the y-axis runs vertically. Each point on a graph is represented by ordered pairs \((x, y)\).

• To graph a line like \( x = -5 \), instead of plotting an equation like \( y = mx + b \), you only need the x-coordinate constant to know the line's position.
• Simply find \( x = -5 \) on the x-axis and draw a vertical line through that point. This line remains parallel to the y-axis and does not cross it.
• Virtual tools or graph paper can help keep such plots neat and accurate.

Graphing helps in visualizing how equations behave and allows one to see relationships between different variables or equations quickly. For vertical lines, it can be an intuitive way to understand their unique properties within a coordinate system.