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Chapter 5: Problem 15

Sketch the line with the given slope and \(y\) -intercept. $$m=0,(0,5)$$

Short Answer

Expert verified
The line is horizontal at \( y = 5 \), through the point (0, 5).

Step by step solution

01

Understand the Equation of the Line

The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. In this problem, \( m = 0 \) and \( b = 5 \), so the equation becomes \( y = 0 \cdot x + 5 \).
02

Simplify the Equation

Since the slope \( m = 0 \), the equation simplifies to \( y = 5 \). This represents a horizontal line, as the value of \( y \) is constant regardless of the value of \( x \).
03

Sketch the Line

To sketch the line, plot the point (0, 5) on the graph, which is the \( y \)-intercept. Since the line is horizontal and \( y = 5 \) for all \( x \), draw a straight, horizontal line through \( y = 5 \). This line is 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.

Equation of a Line
In mathematics, the equation of a line is foundational in understanding linear relationships. Specifically, the slope-intercept form, represented as \( y = mx + b \), is used to describe the line's properties in terms of its slope \( m \) and \( y \)-intercept \( b \). This form is incredibly useful because it readily divulges the steepness and direction of the line (through the slope) and the exact point where the line intersects the y-axis (the y-intercept).
  • The slope \( m \) indicates how much \( y \) changes with respect to \( x \).
  • The \( y \)-intercept \( b \) gives the value of \( y \) where the line crosses the y-axis.
This information allows anyone plotting a graph to quickly sketch the behavior of the line, regardless of its complexity.
Slope
The slope of a line, denoted as \( m \) in the slope-intercept equation, is a measure of its steepness. It tells us how fast \( y \) is changing with respect to \( x \). A positive slope means that as \( x \) increases, \( y \) increases, resulting in an upward-sloping line. Conversely, a negative slope indicates a downward-sloping line, where \( y \) decreases as \( x \) increases. However, in this particular exercise:
  • The slope \( m = 0 \) indicates that the line is completely flat.
  • This signifies no change in the value of \( y \) as \( x \) varies, leading to a horizontal line.
Therefore, in scenarios where the slope is zero, you will encounter horizontal lines parallel to the x-axis.
Y-Intercept
The \( y \)-intercept is a critical point on a graph where a line touches the y-axis. This point describes the value of \( y \) when \( x \) equals zero. In the slope-intercept form \( y = mx + b \), \( b \) is the \( y \)-intercept:
  • It provides an initial condition of the line's position.
  • For this exercise, the \( y \)-intercept is given as 5, meaning the line intersects the y-axis at (0, 5).
With this information, you can quickly pinpoint where to start drawing or analyzing the line's behavior from its intersection at the y-axis, thereby making it a cornerstone of linear graphing.
Horizontal Line
A horizontal line is a special type of line on a graph where the slope is zero. This results in a line that runs parallel to the x-axis, remaining constant as \( x \) changes. As mentioned earlier:
  • The slope \( m = 0 \) simplifies the line's equation to \( y = b \).
  • In our exercise, the equation becomes \( y = 5 \), hence forming a horizontal line.
A striking characteristic of horizontal lines is how all points on the line share the same y-coordinate. Whether \( x \) equals 1, 10, or 100, \( y \) will still be 5 in our case. Recognizing horizontal lines is vital for identifying static relationships in algebra and plotting graphs efficiently.

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