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Chapter 1: Problem 65

Find the slope and \(y\)-intercept of the line, and draw its graph. \(x=3\)

Short Answer

Expert verified
The slope is undefined, and there is no y-intercept.

Step by step solution

01

Identify the type of line

The equation given is in the form of \(x = c\), where \(c\) is a constant. This represents a vertical line passing through the point \((3, y)\) for all values of \(y\).
02

Determine the Slope

For a vertical line, the slope is undefined because the change in \(y\) is not associated with any change in \(x\). Therefore, vertical lines do not have well-defined slopes.
03

Find the y-intercept

A vertical line parallel to the \(y\)-axis does not intersect the \(y\)-axis at any specific point, except if \(x = 0\), which is not the case here.
04

Sketch the Graph

Draw a vertical line on the graph that passes through the \(x\)-coordinate of 3. This line will extend infinitely in both directions along the \(y\)-axis. Since the line is vertical, it will not cross the \(y\)-axis, reflecting that it has no \(y\)-intercept.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope of a Line
The slope of a line represents how steep the line is and is defined as the rate of change of the line along the x-axis and y-axis. However, not all lines have a defined slope. When dealing with vertical lines, such as the one given by the equation \(x=3\), the slope is undefined. This means that for any change in the y-coordinate, the x-coordinate does not change at all, indicating the line goes straight up and down. This is opposite to horizontal lines, where the slope is zero because the y-coordinate does not change while the x-coordinate varies. When thinking about slope in general:
  • Horizontal lines have a slope of zero because they do not rise or fall.
  • Vertical lines have an undefined slope since they involve division by zero (no change in x).
  • Diagonal lines have a defined slope, which can be positive or negative depending on the direction.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. It is found by setting \(x=0\) in the line鈥檚 equation and solving for y. However, vertical lines are special cases. For instance, the equation \(x=3\) describes a line that runs parallel to the y-axis and will never actually intersect it, negating the presence of a y-intercept. This is because the equation never allows for an \(x\)-value of zero, which is needed to calculate where a line meets the y-axis.
  • Vertical lines, like \(x=3\), don't have a y-intercept.
  • To find the y-intercept in general, solve the equation at \(x=0\).
  • Lines that do cross the y-axis will have a clear y-intercept value.
Graphing Lines
Graphing lines involves plotting points on a coordinate plane that satisfy the line鈥檚 equation. For a vertical line, this is straightforward. Given the equation \(x = 3\), it tells us that whatever the y-value might be, the x-value remains 3. This creates a line parallel to the y-axis through \(x = 3\). To graph it:
  • Find the x-coordinate from the equation.
  • Draw a straight vertical line through this x-coordinate.
  • This line will extend infinitely above and below as it does not bend or slope in any direction.
Vertical lines like \(x=3\) will not cross the y-axis, which is why they do not touch any specific y-intercept. Thus, when graphing lines, it鈥檚 crucial to remember that vertical lines will always run parallel to the y-axis, representing constant x-values different from those that make up the y-axis itself.

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