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Chapter 3: Problem 6

Determine the slope of parallel lines and perpendicular lines. $$ y=73 x+32 $$

Short Answer

Expert verified
Parallel slope: 73, Perpendicular slope: \(-\frac{1}{73}\).

Step by step solution

01

Identify the Slope of the Given Line

The equation of the line is given in slope-intercept form, \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m = 73 \).
02

Slope of Parallel Lines

Parallel lines have the same slope. Therefore, the slope of any line parallel to the given line is also \( 73 \).
03

Determine the Slope of Perpendicular Lines

Perpendicular lines have slopes that are negative reciprocals. To find the perpendicular slope, take the negative reciprocal of \( 73 \). The negative reciprocal is \(-\frac{1}{73}\). Hence, the slope of any line perpendicular to the given line is \(-\frac{1}{73}\).

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Key Concepts

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

Parallel Lines
Parallel lines are fascinating in geometry. They have a unique property: they never intersect, no matter how far they are extended in either direction. This is because parallel lines have the same slope. The slope of a line indicates its steepness and direction.
When finding a parallel line to a given line, you simply look at the slope from its equation. For instance, in the equation given by the format \( y = mx + b \), the slope is denoted by \( m \).
In our exercise, the slope of the line is 73.
  • Therefore, the slope of any line parallel to it will also be 73.
  • No additional calculations are necessary, just copy the slope over.
This is a simple yet powerful concept in understanding how lines relate to each other on a graph.
Perpendicular Lines
Perpendicular lines intersect at a right angle (90 degrees). They have an interesting relationship with each other's slopes, which is the key to solving many problems. The slopes of perpendicular lines are negative reciprocals of each other.
To find the negative reciprocal of a slope, flip the fraction upside down and change its sign.
For any slope, say \( m \), its perpendicular counterpart will be \( -\frac{1}{m} \).
  • In the exercise, since the line has a slope of 73, its negative reciprocal is \(-\frac{1}{73}\).
  • This means any line that has this slope \(-\frac{1}{73}\) will be perpendicular to the original line.
Understanding this concept helps in drawing or calculating the positions of lines in geometry.
Slope-Intercept Form
The slope-intercept form is one of the most common ways to represent linear equations. It is written as \( y = mx + b \), where \( m \) denotes the slope and \( b \) represents the y-intercept of the line. In this form, identifying the slope is straightforward since it is right next to the variable \( x \).
This form is particularly useful when you're tasked to quickly determine how a line behaves just by looking at its equation.
  • The slope \( m \) enables you to understand how steep the line is and in which direction it goes (upwards if \( m \) is positive, downwards if \( m \) is negative).
  • The y-intercept \( b \) gives you a head start in graphing as it tells where the line crosses the y-axis.
For the exercise, with an equation \( y = 73x + 32 \), it's clear that the slope \( m \) is 73, making it easy to handle queries about parallel and perpendicular lines.

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