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Chapter 2: Problem 40

Determine if the lines defined by the given equations are parallel, perpendicular, or neither. \(3 y=5\) \(x=1\)

Short Answer

Expert verified
The lines are neither parallel nor perpendicular.

Step by step solution

01

- Convert to Slope-Intercept Form

Rewrite the given equations in the slope-intercept form, which is given by \( y = mx + b \), where \( m \) is the slope.
02

- Simplify the First Equation

The first equation is \( 3y = 5 \). Divide both sides by 3 to solve for \( y \): \[ y = \frac{5}{3} \]. This equation can be written as \( y = 0x + \frac{5}{3} \), where the slope \( m = 0 \).
03

- Identify the Slope of Second Equation

The second equation is \( x = 1 \). This represents a vertical line. A vertical line has an undefined slope.
04

- Compare Slopes

Parallel lines have the same slope, and perpendicular lines have slopes that are negative reciprocals of each other. Since the first line has a slope of 0 and the second line has an undefined slope, they are neither parallel nor perpendicular.

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Key Concepts

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

slope-intercept form

Understanding the slope-intercept form is crucial in algebra. The slope-intercept form of a line is:

\( y = mx + b \),
where:
  • \( y \) represents the y-coordinate
  • \( m \) is the slope of the line
  • \( x \) is the x-coordinate
  • \( b \) is the y-intercept
The slope \( m \) indicates the steepness of the line, and the y-intercept \( b \) is where the line crosses the y-axis.

For example, the equation \( y = \frac{5}{3} \) doesn't have an \( x \) term, so it can be thought of as \( y = 0x + \frac{5}{3} \). Here, the slope \( m = 0 \).
parallel lines

Parallel lines are lines that never intersect.
To determine if lines are parallel, they must have the same slope but different y-intercepts.
For example, lines given by the equations \( y = 2x + 3 \) and \( y = 2x - 4 \) are parallel because their slopes are both \( 2 \).
In the current exercise, the first line has a slope of \( 0 \) and the second line has an undefined slope. Hence, they are not parallel.
perpendicular lines

Perpendicular lines intersect at a right angle (90掳).
These lines have slopes that are negative reciprocals of each other.
If line 1 has a slope of \( m \), then line 2 must have a slope of \( -\frac{1}{m} \) for them to be perpendicular.
For instance, a line with a slope of \( 2 \) is perpendicular to a line with a slope of \( -1/2 \).
In our exercise, the first line has a slope of \( 0 \) and the second line has an undefined slope. They cannot be perpendicular since the negative reciprocal of \( 0 \) is undefined.
undefined slope

An undefined slope means the line is vertical.
In an equation like \( x = 1 \), the slope cannot be defined.
The line runs straight up and down and doesn't cross the y-axis.
Unlike horizontal lines, these lines don't have an equation in y = mx+b form.
They are crucial in understanding the special cases in coordinate geometry.
slope

The slope of a line measures its steepness.
It's calculated as the change in y divided by the change in x, given by:\[ m = \frac{\Delta y}{\Delta x} \]
Key characteristics:
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves from left to right.
  • A slope of zero means the line is horizontal.
  • An undefined slope means the line is vertical.
Understanding slope helps in identifying whether lines are parallel, perpendicular, or neither.

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Most popular questions from this chapter

Refer to the functions \(r, p,\) and \(q .\) Evaluate the function and write the domain in interval notation. \(r(x)=-3 x \quad p(x)=x^{2}+3 x \quad q(x)=\sqrt{1-x}\) $$\left(\frac{p}{q}\right)(x)$$

Given \(g(x)=\sqrt{x-3}\), evaluate \((g \circ g)(x)\) and write the domain in interval notation.

Determine if the function is even, odd, or neither. $$ f(x)=3 x^{6}+2 x^{2}+|x| $$

Graph the function. $$ h(x)=\left\\{\begin{array}{ll} -2 x & \text { for } x<0 \\ \sqrt{x} & \text { for } x \geq 0 \end{array}\right. $$

a. Given \(n(x)=7|x|+3 x-1,\) find \(n(-x)\). b. Find \(-n(x)\). c. Is \(n(-x)=n(x)\) ? d. Is \(n(-x)=-n(x)\) ? e. Is this function even, odd, or neither?
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