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Chapter 12: Problem 46

Find the equation of the line through the given points. $$(-3,-4) \text { and }(5,-4)$$

Short Answer

Expert verified
The equation of the line through the points (-3,-4) and (5,-4) is \(y = -4\).

Step by step solution

01

Identify common characteristics of the points

Review the given points, (-3,-4) and (5,-4). They share the same y-coordinate, -4, which means that the line passing through these points is a horizontal line.
02

Determine the slope and y-intercept

The slope, m, of a horizontal line is 0. The y-intercept of the line is equal to the y-coordinate of any point on the line. Since both points on the line have y-coordinate -4, the y-intercept, b, is -4.
03

Write the equation of the line

Now we can write the equation of the line in slope-intercept form, y = mx + b, substituting the slope and y-intercept we found earlier. Thus, the equation of the line is y = 0x - 4.
04

Simplify the equation

Simplify the equation further. Since 0x equals 0, we can eliminate this part of the equation. As a result, the final equation of the line is y = -4.

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Key Concepts

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

Horizontal Line
A horizontal line is one of the simplest types of lines in geometry. It runs from left to right and is parallel to the x-axis in a coordinate plane.
  • The most distinctive feature of a horizontal line is that it has no vertical change. All the y-coordinates of points on a horizontal line are identical.
  • In the given exercise, the y-coordinate is -4 for both points e{(-3, -4)} and e{(5, -4)}, confirming that the line is indeed horizontal.
A horizontal line can be easily recognized by this constant y-value. No matter how far you move along it to the left or the right, its height remains unchanged.
Slope-Intercept Form
The slope-intercept form is a way of writing the equation of a line. It is represented as: \[ y = mx + b \]
  • Here, \( m \) represents the slope, which signifies the steepness or angle of the line.
  • The \( b \) is the y-intercept, which specifies where the line crosses the y-axis.
This form is particularly useful because it provides immediate information about the line's characteristics like the slope and y-intercept. In the exercise, despite having a zero slope due to the horizontal nature of the line, the slope-intercept form refreshes our memory on how lines fit within this expressive framework.
Y-Intercept
The y-intercept of a line is where it crosses the y-axis. In our equation of the line, this is represented by the term \( b \). For a horizontal line:
  • This intercept is simply the constant y-value of the line.
  • It tells us the vertical position of the line on the graph.
In the exercise, because the y-coordinate is e{-4}, the y-intercept is also e{-4}. Regardless of the line's length, when a line is horizontal, its y-intercept is a critical point for defining its equation.
Slope
The slope of a line, indicated by \( m \), measures how steep the line is. It reflects the rate at which the y-coordinate changes with respect to the x-coordinate. With a horizontal line:
  • The slope is always 0 because there is no rise over run; the line does not rise or fall.
  • This specifically means that for any two points on the line, the difference in y-coordinates is zero.
In the original exercise, both points e{(-3, -4)} and e{(5, -4)} showcase a slope of zero. This characteristic reinforces the idea that horizontal lines reflect zero decline or incline, simplifying their representation in slope-intercept form.

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Most popular questions from this chapter

According to some veterinarians, the age \(x\) of a dog can be translated to "human years" by using the equation \(H=4 x+16,\) where \(H\) is the human equivalent age for the dog. Graph this equation for values of \(x\) from 2 to \(21 .\) The point whose coordinates are \((6,40)\) is on the graph. Write a sentence that explains the meaning of this ordered pair.

Evaluate a function. Given \(s(t)=\frac{3}{t-1},\) find \(s(-2)\).

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(5,1), P_{2}(3,-2) ; Q_{1}(0,-2), Q_{2}(3,-4)$$

What effect does decreasing the coefficient of \(x\) have on the graph of \(y=m x+b\) \(m>0 ?\)

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(0,1), P_{2}(2,4) ; Q_{1}(-4,-7), Q_{2}(2,5)$$
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