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Chapter 4: Problem 353

Use slopes and \(y\) -intercepts to determine if the lines are parallel. \(x=7 ; \quad x=-8\)

Short Answer

Expert verified
The lines are parallel because both have undefined slopes.

Step by step solution

01

Identify the Equations

Given the two equations: 1) \(x = 7\)2) \(x = -8\)
02

Interpret the Equations

Both equations represent vertical lines. A vertical line has an undefined slope and passes through all points with the given x-coordinate.
03

Determine the Slopes

For vertical lines \(x = 7\) and \(x = -8\), the slopes are undefined for both.
04

Compare Slopes and Determine if Parallel

Since both lines have undefined slopes and the same slope, they are parallel to each other by definition.

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Key Concepts

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

Slope and Its Importance
When learning about lines in geometry and algebra, the slope is a crucial concept. Slope measures the steepness of a line and is calculated as the ratio of the 'rise' (change in y) over the 'run' (change in x). Symbolically, this is represented as: \[ m = \frac{\Delta y}{\Delta x} \]. If the slope is positive, the line rises as it moves to the right. If negative, the line falls as it moves to the right. A zero slope means the line is horizontal, and if neither x nor y changes, the slope is undefined.
Vertical Lines
Vertical lines are special cases in geometry. Unlike typical lines, they do not have a 'run'; they go straight up and down. This means that their slope is undefined. Every vertical line can be written in the form x = a, where a is the x-coordinate for every point on the line. These lines are parallel to each other because they all have the same undefined slope. For example, the lines given by the equations x = 7 and x = -8 in our problem are vertical lines and are, therefore, parallel.
Understanding Undefined Slope
When we talk about 'undefined slope,' we mean the slope does not fit into the normal ratio of rise over run. In mathematical terms, this happens because the run (change in x) is zero. Dividing by zero is undefined in mathematics, hence the term 'undefined slope.' Vertical lines always have an undefined slope because they do not 'move' horizontally. Remember, any two vertical lines will be parallel to each other as they both have this undefined slope.

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Most popular questions from this chapter

Find the equation of each line. Write the equation in slope-intercept form. Parallel to the line \(2 x+3 y=6\), containing point (0,5)

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