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Chapter 8: Problem 64

How can you tell by inspection that \(y=2 x-4\) and \(y=-3 x-1\) are not parallel lines?

Short Answer

Expert verified
The lines are not parallel because their slopes are different.

Step by step solution

01

Identify the Equation Forms

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

Determine the Slope of Each Line

For the line equation \(y = 2x - 4\), the slope \(m_1\) is 2. For the line \(y = -3x - 1\), the slope \(m_2\) is -3.
03

Compare the Slopes

Two lines are parallel if their slopes are equal. Here, \(m_1 = 2\) is not equal to \(m_2 = -3\).

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Key Concepts

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

Slope-Intercept Form
In algebra, the slope-intercept form is a way of writing the equation of a line so you can easily identify its slope and y-intercept. This form is written as:
  • \( y = mx + b \)
  • \( m \) represents the slope of the line
  • \( b \) is the y-intercept, the point where the line crosses the y-axis
The main idea is that the slope \( m \) tells you how steep the line is. Meanwhile, \( b \) gives you the exact point where the line touches the y-axis.
For example, in the equation \( y = 2x - 4 \), the slope \( m \) is 2, and the y-intercept \( b \) is -4. This means the line rises 2 units for every 1 unit it moves to the right.
Equation Forms
Understanding different forms of an equation is crucial because they give you specific information about the line. The slope-intercept form is just one of several ways to express a linear equation. Here are a few others you might encounter:
  • Point-slope form: \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a known point on the line.
  • Standard form: \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be non-negative.
Each form has its own uses, with slope-intercept being particularly handy for quickly determining the line's slope and y-intercept.
Choosing the right form can make solving, graphing, and understanding equations much easier.
Comparing Slopes
The slope of a line is a measure of its steepness, and it's what we compare to see if two lines are parallel. Remember, parallel lines never meet and have equal slopes.
To compare slopes:
  • Look at the coefficient of \( x \) in each equation, which gives the slope.
  • For example, in \( y = 2x - 4 \), the slope is 2.
  • In \( y = -3x - 1 \), the slope is -3.
  • If two slopes are equal, the lines are parallel.
  • If they're different, like 2 and -3, the lines are not parallel.
In the given exercise, because the slopes are not equal, \( y = 2x - 4 \) and \( y = -3x - 1 \) are not parallel lines. This concept is crucial in geometry and helps us understand relationships between different lines.

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Most popular questions from this chapter

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ 9 x+7 y=0 $$

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point \((-1,4)\) and is parallel to the line \(x-2 y=6\)

Contains the origin and is perpendicular to the line \(-2 x+3 y=8\)

Determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. (Objective 2) $$-2 x-11 y=11$$

Find the equation of the line that contains the given point and has the given slope. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. (Objective 1a) $$(-3,9), m=0$$
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