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Chapter 1: Problem 102

True or False? In Exercises 101 and \(102,\) determine whether the statement is true or false. Justify your answer. $$\begin{array}{l}{\text { The line through }(-8,2) \text { and }(-1,4) \text { and the line }} \\ {\text { through }(0,-4) \text { and }(-7,7) \text { are parallel. }}\end{array}$$

Short Answer

Expert verified
False. The line through (-8,2) and (-1,4) and the line through (0,-4) and (-7,7) are not parallel.

Step by step solution

01

Calculate the slope of the first line

The slope of a line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated as \[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \] So, for the points (-8,2) and (-1,4), the slope is \[ m_1 = \frac{(4 - 2)}{(-1 - -8)} = \frac{2}{7} \]
02

Calculate the slope of the second line

Using the same formula, the slope of the line through the points (0,-4) and (-7,7) is \[ m_2 = \frac{(7 - -4)}{(-7 - 0)} = \frac{11}{-7} = -\frac{11}{7} \]
03

Compare the slopes of both lines

Since the slopes \( m_1 \) and \( m_2 \) are not equal, the lines are not parallel.

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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 lines in a plane that never meet. Regardless of how far they are extended, they always remain the same distance apart. The mathematical way to determine if two lines are parallel is by evaluating their slopes.
For two lines to be parallel, their slopes must be equal. Essentially, if line 1 has a slope \( m_1 \) and line 2 has a slope \( m_2 \), then these lines are parallel if \( m_1 = m_2 \).
In the context of coordinate geometry, this means that the ratio of the vertical change to the horizontal change between any two points on the lines must be identical for both lines.
  • This constancy ensures the lines never converge or diverge.
Slope of a line
The slope of a line is a measure of its steepness. It indicates how much the line rises or falls as you move along it. Mathematically, the slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is defined as
\[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \]
This formula describes the change in \( y \) (vertical change) over the change in \( x \) (horizontal change).
A positive slope implies the line is rising, while a negative slope indicates the line is falling. Zero slope signifies a horizontal line. If the slope is undefined, the line is vertical.
  • In the exercise, for the first line through points \((-8,2)\) and \((-1,4)\), the slope is \( \frac{2}{7} \).
  • For the second line through points \((0,-4)\) and \((-7,7)\), the slope is \(-\frac{11}{7} \).
Equation of a line
The equation of a line is a mathematical representation showing the relationship between \( x \) and \( y \) coordinates on a two-dimensional plane. One common form of a linear equation is the slope-intercept form
\[ y = mx + c \]
Here, \( m \) represents the slope, and \( c \) is the y-intercept - the point where the line crosses the y-axis.
Given two points on a line, one can find the equation by first determining the slope and then using either point to solve for \( c \).
  • This helps in constructing a full description of the line in its graphical form.
  • It can be extended to find parallel or perpendicular lines through transformations of \( m \) and \( c \).
Coordinate geometry
Coordinate geometry, also known as analytic geometry, explores geometric problems using coordinates and algebra. It transforms shapes and figures into algebraic expressions through coordinate points.
Using coordinate geometry, one can easily determine properties like distance, midpoint, slope, and relationships between geometric figures.
  • For instance, coordinate geometry simplifies the problem of understanding if lines are parallel by using the slope criterion.
  • It gives a powerful connection between algebra and geometry.
  • By plotting points and drawing the lines, visual understanding of the concepts is enhanced.

This blend of algebraic and geometric techniques offers a robust way to solve complex problems in a straightforward manner.

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Most popular questions from this chapter

Graphical Reasoning Graph each of the functions with a graphing utility. Determine whether the function is even, odd, or neither. $$\begin{array}{ll}{f(x)=x^{2}-x^{4}} & {g(x)=2 x^{3}+1} \\ {h(x)=x^{5}-2 x^{3}+x} & {j(x)=2-x^{6}-x^{8}} \\ {k(x)=x^{5}-2 x^{4}+x-2} & {p(x)=x^{9}+3 x^{5}-x^{3}+x}\end{array}$$ What do you notice about the equations of functions that are odd? What do you notice about the equations of functions that are even? Can you describe a way to identify a function as odd or even by inspecting the equation? Can you describe a way to identify a function as neither odd nor even by inspecting the equation?

True or False? In Exercises 95 and \(96,\) determine whether the statement is true or false. Justify your answer. If the inverse function of \(f\) exists and the graph of \(f\) has a \(y\) -intercept, then the \(y\) -intercept of \(f\) is an \(x\) -intercept of \(f^{-1}\) .

Proof Prove that if \(f\) and \(g\) are one-to-one functions, then \((f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x)\)

Describing Profits Management originally predicted that the profits from the sales of a new product would be approximated by the graph of the function \(f\) shown. The actual profits are shown by the function \(g\) along with a verbal description. Use the concepts of transformations of graphs to write \(g\) in terms of \(f .\) (a) The profits were only three-fourths as large as expected. (b) The profits were consistently \(\$ 10,000\) greater than predicted. (c) There was a two-year delay in the introduction of the product. After sales began, profits grew as expected.

Comparing Slopes Use a graphing utility to compare the slopes of the lines \(y=m x\) , where \(m=0.5,1,2,\) and \(4 .\) Which line rises most quickly?Now, let \(m=-0.5,-1,-2,\) and \(-4 .\) Which line falls most quickly? Use a square setting to obtain a true geometric perspective. What can you conclude about the slope and the "rate" at which the line rises or falls?
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