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Chapter 3: Problem 138

In the following exercises, use slopes and \(y\) -intercepts to determine if the lines are parallel, perpendicular, or neither. $$ x=-3 ; \quad x=-2 $$

Short Answer

Expert verified
The lines are parallel.

Step by step solution

01

Identify Slopes of the Lines

First, recognize that the given equations are in the form of vertical lines. A vertical line has the equation of the form: x = a. Therefore, the lines given in the problem are vertical lines at different x-values.
02

Determine Slopes of Vertical Lines

Recall that the slope of a vertical line is undefined, because the change in x is zero while the change in y can be any non-zero number.
03

Compare the Lines

Vertical lines are parallel to each other because their slopes cannot be calculated. Therefore, they run in the same direction and do not intersect.
04

Conclusion

Since both lines x = -3 and x = -2 are vertical, they are parallel.

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Key Concepts

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

slopes of lines
One of the most essential concepts to understand when dealing with lines is their slope. The slope essentially measures how steep a line is, or how much the y-value changes for a given change in the x-value. In mathematical terms, the slope (denoted as m) is calculated as the ratio of the rise (the change in y) to the run (the change in x). It is given by the formula: \( m = \frac{\Delta y}{\Delta x} \). This formula helps us determine whether lines are increasing, decreasing, or constant as we move from left to right.
It鈥檚 important to know:
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it moves from left to right.
  • If the slope is zero, the line is horizontal.
  • Vertical lines, however, have an undefined slope because their x-values don't change, leading to a zero in the denominator of our slope formula.
Knowing the slope helps in determining the relationship between lines such as being parallel or perpendicular.
y-intercepts
The y-intercept of a line is the point where the line crosses the y-axis. This is represented as a point in the format (0, b), where b is the value of the y-intercept. The slope-intercept form of a line鈥檚 equation, which is y = mx + b, clearly shows how the slope (m) and y-intercept (b) interact to form the line.
Some key points to note:
  • The y-intercept tells us where the line will start on the y-axis when x is zero.
  • In practical problems, knowing the y-intercept can help plot the line after identifying the slope.
  • When comparing two lines in terms of their y-intercepts, if they are different, the lines will cross the y-axis at different points; otherwise, they will cross at the same point.
A clear understanding of y-intercepts is crucial when graphing equations or determining line relationships.
vertical lines
Vertical lines are a special type of line in geometry and algebra. Unlike most lines which are defined with the equation y = mx + b, vertical lines are defined by the equation x = a where 'a' is the x-coordinate where the line crosses the x-axis.
Here's what you need to understand about vertical lines:
  • Vertical lines have an undefined slope because the denominator in the slope formula (change in x) equals zero.
  • These lines run parallel to the y-axis and never intersect it, meaning they cover all y-values at a specific x-value.
  • In the context of comparing lines, any two vertical lines are always parallel to each other as they run in the same direction along the y-axis.
A great example of vertical lines in the exercise provided are x=-3 and x=-2, which are parallel vertical lines, exemplifying the nature of vertical lines aptly.

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