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Chapter 2: Problem 92

Find the slope and y-intercept of each line. Graph the line. $$ y=-1 $$

Short Answer

Expert verified
Slope: 0, Y-intercept: -1

Step by step solution

01

Identify the Equation Type

Notice that the equation given is in the form of a horizontal line, where the equation is simply y = constant.
02

Determine the Slope

For a horizontal line, the slope is always zero. Thus, in this case, the slope (m) is 0.
03

Determine the Y-Intercept

The given equation is y = -1, which means that the line intersects the y-axis at -1. Therefore, the y-intercept is -1.
04

Graph the Line

To graph the line y = -1, draw a horizontal line that crosses the y-axis at -1. This line will be parallel to the x-axis.

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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 a straight line that has the same y-coordinate for every point on the line. This means that no matter what the x-coordinate is, the value of y remains constant.
For example, the equation of a horizontal line can be written as y = c, where c is a constant. If c equals -1, the equation becomes y = -1.
In this case, every point on the line will have a y-coordinate of -1. No matter how far you move along the x-axis, the y-coordinate will always be -1.
This characteristic of horizontal lines makes them quite easy to recognize and graph.
To sum up:
  • A horizontal line has a constant y-coordinate.
  • The equation of a horizontal line is in the form y = c.
  • It is parallel to the x-axis.
slope
The slope of a line indicates its steepness and direction. It is calculated as the 'rise over run,' which means the change in y over the change in x.
Mathematically, the slope (m) is given by:
\(m = \frac{\Delta y}{\Delta x}\).
For a horizontal line, however, the change in y (abla y) is zero since all y-coordinates are the same.
Therefore, for a horizontal line: \(m = \frac{0}{\Delta x} = 0\).
A zero slope means the line is perfectly flat, with no incline or decline.
In summary:
  • The slope formula is \(m = \frac{\Delta y}{\Delta x}\).
  • For horizontal lines, the slope is always 0.
  • A zero slope indicates a lack of incline.
y-intercept
The y-intercept is where the line crosses the y-axis. It indicates the value of y when x is zero.
You can find the y-intercept by setting x to zero in the equation of the line.
For example, in the equation y = -1, the y-intercept is -1 because the line crosses the y-axis at this point.
The y-intercept is a crucial piece of information that helps you to accurately graph a line.
To recap:
  • The y-intercept is the y-value when x is zero.
  • For the equation y = -1, the y-intercept is -1.
  • The y-intercept helps you graph the line more effectively.
graphing linear equations
Graphing linear equations involves plotting points that satisfy the equation and then drawing a line through these points.
For a horizontal line like y = -1, you only need to know that all y-values are -1.
Start by plotting a series of points where the y-coordinate is -1, such as (0, -1), (1, -1), (2, -1), and so on.
Connect these points with a straight line that is parallel to the x-axis.
This is a very straightforward process:
  • Plot points that satisfy the equation.
  • Connect the points with a straight line.
  • For y = -1, the line is parallel to the x-axis.
By following these steps, you can easily graph any linear equation, including horizontal lines.

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Most popular questions from this chapter

The horsepower (hp) that a shaft can safely transmit varies directly with its speed (in revolutions per minute, rpm) and the cube of its diameter. If a shaft of a certain material 2 inches in diameter can transmit 36 hp at \(75 \mathrm{rpm},\) what diameter must the shaft have in order to transmit 45 hp at 125 rpm?

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