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Chapter 0: Problem 17

Graph. List the slope and \(y\) -intercept. $$y=7$$

Short Answer

Expert verified
Slope: 0, y-intercept: 7

Step by step solution

01

Identify the form of the equation

The given equation is in the form of slope-intercept form, which is generally written as \[ y = mx + b \]. In this case, the equation is \[ y = 7 \].
02

Identify the slope (m)

From the equation \[ y = 7 \], there is no \( x \) term present. This implies that the slope \( m \) is 0 because the equation can be seen as \[ y = 0x + 7 \].
03

Identify the y-intercept (b)

The \( y \)-intercept can be identified directly from the equation \[ y = 7 \]. Here, the \( y \)-intercept \( b \) is 7.
04

Summarize the results

Therefore, the slope \( m \) is 0 and the \( y \)-intercept \( b \) is 7. The equation of the line represents a horizontal line crossing the \( y \)-axis at 7.

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Key Concepts

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

slope
The concept of the slope is fundamental in understanding the behavior of linear equations in the form \( y = mx + b \). The slope, represented by \(m\), determines how steep a line is and its direction: whether it rises or falls as you move from left to right.
When looking at an equation like \(y = 7\), we notice that there is no \(x\) term. This scenario tells us that the slope \(m\) equals 0. A slope of 0 means that there is no vertical change as you move along the line鈥攅ssentially, the line remains perfectly flat or horizontal.
In summary:
  • The slope \(m\) tells you the steepness and direction of the line.
  • A slope of 0 indicates a horizontal line.
  • For the equation \(y = 7\), the slope \(m\) is 0.
y-intercept
The \(y\)-intercept is the point where a line crosses the \(y\)-axis. It is represented by the value \(b\) in the slope-intercept form of a linear equation, \(y = mx + b\).
In the equation \(y = 7\), there is no \(x\) term, which means that the line crosses the \(y\)-axis at the point where \(y = 7\). Thus, the \(y\)-intercept is 7. This concept is essential in graphing because it tells you where to draw the line on the \(y\)-axis.
To summarize:
  • The \(y\)-intercept \(b\) tells you where the line crosses the \(y\)-axis.
  • For the equation \(y = 7\), the \(y\)-intercept \(b\) is 7.
horizontal line
A horizontal line is a straight line that runs from left to right and is parallel to the \(x\)-axis. When dealing with linear equations, a horizontal line has a slope (\(m\)) of 0, meaning there is no vertical change as you move along the line.
In the given equation \(y = 7\), the line is horizontal because the slope is 0. This line crosses the \(y\)-axis at 7 and remains parallel to the \(x\)-axis as you move along it. It means that all points on this line have their \(y\)-coordinate equal to 7, while the \(x\)-coordinate can vary.
To summarize:
  • A horizontal line has a slope (\(m\)) of 0.
  • The line remains parallel to the \(x\)-axis.
  • For the equation \(y = 7\), the line is horizontal and crosses the \(y\)-axis at 7.

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