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Chapter 3: Problem 12

Determine the slope of parallel lines and perpendicular lines. $$ x+y=0 $$

Short Answer

Expert verified
Slope of parallel lines: -1; Slope of perpendicular lines: 1.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The given equation is \( x + y = 0 \). To find the slope, we need to rewrite it in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. Subtract \( x \) from both sides: \( y = -x \). This simplifies to \( y = -1x + 0 \). Thus, the slope \( m = -1 \).
02

Determine the Slope of Parallel Lines

Lines that are parallel to each other have the same slope. Since the slope of the given line is \(-1\), all lines parallel to it also have a slope of \(-1\).
03

Calculate the Slope of Perpendicular Lines

To find the slope of lines perpendicular to a given line, take the negative reciprocal of the given slope. The slope of the original line is \(-1\); thus, the negative reciprocal of \(-1\) is \(1\). Therefore, the slope of lines perpendicular to the original line is \(1\).

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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 those lines in a plane that never intersect. They always maintain the same distance from each other. A key characteristic of parallel lines is that they have the same slope. This means that two parallel lines rise and run identically. For example, if one line has a slope of \-1\, then any line parallel to it will also have a slope of \ -1 \. This consistent slope ensures that they remain equidistant no matter how far they extend in either direction.
  • The slope of a line is a measure of its steepness.
  • Two parallel lines will have identical slopes.
  • For the equation \( x + y = 0 \), rewritten as \( y = -x\), the slope is \( -1 \).
Remember, parallelism in lines is all about having the same slope, which guarantees they will never meet.
Perpendicular Lines
Perpendicular lines intersect at a right angle (90 degrees). The slopes of these lines have a unique relationship: they are negative reciprocals of each other. This special characteristic ensures that the lines meet at a perfect right angle.
To calculate the slope of a perpendicular line, you take the original line鈥檚 slope and find its negative reciprocal. In mathematical terms, if a line has a slope \(m\), a line perpendicular to it will have a slope of \(-1/m\).
  • Use negative reciprocals to find slopes of perpendicular lines.
  • For a line with a slope of \(-1\), the perpendicular line鈥檚 slope will be \(1\).
  • This relationship creates the 90-degree intersection.
So, if you're given a line, you can quickly determine the slope of a line perpendicular to it by applying this reciprocal rule.
Slope-Intercept Form
The slope-intercept form is a way of writing a linear equation to easily identify both the slope and the y-intercept of a line. The standard form of this equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
This form is especially useful in graphing because:
  • The slope \(m\) tells us how steep the line is and the direction it moves.
  • The y-intercept \(b\) gives the point where the line crosses the y-axis.
For example, with the equation \(y = -x\), we can see the slope is \(-1\) and the y-intercept is \(0\).
Keep in mind that converting an equation to slope-intercept form can provide a quicker and clearer understanding of the line鈥檚 properties, making graphing and comparison more intuitive.

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