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Chapter 1: Problem 24

In each part, classify the lines as parallel, perpendicular, or neither. (a) \(y=-5 x+1\) and \(y=3-5 x\) (b) \(y-1=2(x-3)\) and \(y-4=-\frac{1}{2}(x+7)\) (c) \(4 x+5 y+7=0\) and \(5 x-4 y+9=0\) (d) \(A x+B y+C=0\) and \(A x+B y+D=0\) (e) \(y=\frac{1}{2} x\) and \(x=\frac{1}{2} y\)

Short Answer

Expert verified
(a) Parallel; (b) Perpendicular; (c) Perpendicular; (d) Parallel; (e) Neither.

Step by step solution

01

Identify Slope of Each Line

The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) is the slope. For lines given in point-slope form or standard form, convert to slope-intercept form to find the slope. The slopes of parallel lines are equal, while the product of the slopes of perpendicular lines is -1.
02

Evaluate Part (a)

Both lines are in slope-intercept form: \( y = -5x + 1 \) and \( y = -5x + 3 \). The slope \( m \) of each line is \(-5\). Since the slopes are equal, the lines are parallel.
03

Evaluate Part (b)

Convert both equations to slope-intercept form. For \( y - 1 = 2(x - 3) \), simplify to get \( y = 2x - 5 \), with slope \( m = 2 \). For \( y - 4 = -\frac{1}{2}(x + 7) \), simplify to get \( y = -\frac{1}{2}x + \frac{7}{2} + 4 \) and further simplify to \( y = -\frac{1}{2}x + \frac{15}{2} \), with slope \( m = -\frac{1}{2} \). The product \( 2 \times -\frac{1}{2} = -1 \) indicates the lines are perpendicular.
04

Evaluate Part (c)

First, rewrite both equations in slope-intercept form. For \( 4x + 5y + 7 = 0 \), simplify to get \( y = -\frac{4}{5}x - \frac{7}{5} \), giving a slope of \(-\frac{4}{5} \). For \( 5x - 4y + 9 = 0 \), simplify to \( y = \frac{5}{4}x + \frac{9}{4} \), giving a slope of \( \frac{5}{4} \). The product \(-\frac{4}{5} \times \frac{5}{4} = -1 \), so the lines are perpendicular.
05

Evaluate Part (d)

Both equations are in the form \( Ax + By + C = 0 \) and \( Ax + By + D = 0 \). Since the coefficients of \( x \) and \( y \) are identical, the lines have the same slope and are parallel unless \( C = D \), whereby they are identical lines.
06

Evaluate Part (e)

The first line \( y = \frac{1}{2}x \) is already in slope-intercept form with a slope of \( \frac{1}{2} \). The second line \( x = \frac{1}{2}y \) can be rewritten as \( y = 2x \), giving a slope of \( 2 \). The product \( \frac{1}{2} \times 2 = 1 \) shows they are neither parallel nor perpendicular as the product is not -1.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is a popular way to express the equation of a straight line. It is given as \(y = mx + b\), where:
  • \(m\) represents the slope of the line.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
This form is particularly useful because it provides immediate information about both the slope and the starting point on the y-axis. If you know these two values, you can graph the line quickly. You simply start at the y-intercept on the y-axis and use the slope \(m\) to determine the angle and direction of the line as it ascends or descends across the graph.
Standard Form
The standard form of a linear equation is \(Ax + By = C\). This format can sometimes make solving certain types of problems easier, particularly those involving systems of equations. Here, \(A\), \(B\), and \(C\) are integers, and \(A\) should typically be positive.
  • Lines in standard form can be turned into slope-intercept form by solving for \(y\). For example, \(4x + 5y = 20\) could be rearranged to \(y = -\frac{4}{5}x + 4\).
  • The form allows easy determination of both x- and y-intercepts by setting either variable to zero.
Using the standard form, we can easily compare coefficients when determining the parallelism or perpendicularity of two lines.
Point-Slope Form
Point-slope form is another way to express the equation of a straight line and is particularly helpful when you know one point on the line and the slope. The formula is written as \(y - y_1 = m(x - x_1)\), where:
  • \(m\) is the slope of the line.
  • \((x_1, y_1)\) is a known point on the line.
From this form, it's straightforward to convert to slope-intercept form. Simply solve the equation for \(y\) to reorganize it into \(y = mx + b\). The point-slope form is incredibly convenient when you need to quickly write an equation knowing just a point and a slope, without requiring the y-intercept directly.
Slope Calculation
The slope of a line measures its steepness and direction. It can be calculated when two points on the line are known using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). The slope \(m\) is:
  • Positive if the line ascends from left to right.
  • Negative if the line descends from left to right.
  • Zero if the line is horizontal, indicating no change in y-values as x-values increase.
  • Undefined for vertical lines, since division by zero occurs \((x_2 - x_1 = 0)\).
Knowing how to calculate a slope is critical for classifying lines as parallel or perpendicular since parallel lines share the same slope, and the slopes of perpendicular lines multiply to \(-1\).
Linear Equations
Linear equations form the basis for straight lines on a graph and can be expressed in multiple forms including slope-intercept, standard, and point-slope. The presence of a linear equation indicates a constant rate of change, signifying that the graph's inclination remains consistent throughout.
  • They simplify the determination of relationships between variables.
  • Graphically, each solution of the equation corresponds to a point on the line.
Linear equations are also central in real-world applications where relationships between two quantities can be tracked steadily, such as speed over time or costs with varying production levels. Understanding and transforming between different forms of linear equations allow for adaptability across various contexts.

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