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Chapter 5: Problem 27

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

Short Answer

Expert verified
The line is horizontal at \(y = 5\).

Step by step solution

01

Identify Line Components

For a line equation of the form \(y = mx + b\), \(m\) represents the slope and \(b\) represents the y-intercept. Here, the slope \(m = 0\) and the y-intercept is \(5\), which means the line passes through the point \((0, 5)\).
02

Understand the Slope

A slope \(m = 0\) indicates that the line is horizontal. A horizontal line means that as we move along the line, \(y\) does not change; it stays constant no matter the value of \(x\).
03

Draw the Y-Intercept

Start by plotting the y-intercept on a coordinate plane. Since the y-intercept is \(5\), plot the point \((0, 5)\) on the y-axis.
04

Sketch the Line

Draw a horizontal line through the point \((0, 5)\). Since the slope is zero, this line stays 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.

Understanding Slope
The concept of slope is crucial in understanding how a line behaves on a coordinate plane. Slope, often denoted by the letter \(m\), measures the steepness or inclination of a line. It is calculated by the ratio of the "rise" (the change in \(y\)-values) to the "run" (the change in \(x\)-values).

For a line equation in the form \(y = mx + b\), the slope \(m\) tells us how much \(y\) changes for every change in \(x\).
  • If the slope is positive, the line goes upwards as you move from left to right.
  • If the slope is negative, the line descends as you move from left to right.
  • A slope of zero indicates a horizontal line, meaning there is no vertical change as \(x\) changes.
In the exercise example, the slope is zero, resulting in a perfectly horizontal line. This means regardless of how much we move left or right from a point, the line remains at the same \(y\)-value.
The Role of the Y-Intercept
The y-intercept is another essential component in the line equation \(y = mx + b\). It is represented by \(b\), the constant part of the equation, and indicates where the line crosses the y-axis. This point is where \(x = 0\).

To find the y-intercept, simply set \(x = 0\) and solve for \(y\), resulting in \(y = b\). In our exercise, the y-intercept is \(5\), meaning the line intersects the y-axis at the point \((0, 5)\).
  • The y-intercept provides a starting point for graphing the line.
  • It is crucial for drawing the line accurately on a coordinate plane.
  • Even when the slope changes, this point remains constant, anchoring the line.
Knowing the y-intercept helps in visualizing the line's position without needing to plot multiple points.
Navigating the Coordinate Plane
A coordinate plane is a two-dimensional surface where we can plot points, lines, and curves using a pair of axes: the x-axis (horizontal) and the y-axis (vertical).

It is structured as a grid and is essential for graphing linear equations. Each point on this plane corresponds to a pair of numbers \((x, y)\), which helps locate its position.
  • The x-axis measures horizontal displacement.
  • The y-axis measures vertical displacement.
  • Origin, the point \((0, 0)\), is where both axes intersect and is the starting point for measuring everything on the plane.
In the context of our exercise, plotting a line involves:

1. Identifying the y-intercept point \((0, 5)\).
2. Understanding that a horizontal slope means drawing a line parallel to the x-axis.

This method of visualization ensures that you plot lines accurately, respecting both the slope and y-intercept instructions.

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