/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 46 Graph each linear equation. See ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 46

Graph each linear equation. See Examples 1 through 7. $$ y-6=0 $$

Short Answer

Expert verified
Graph y = 6 as a horizontal line.

Step by step solution

01

Rewrite the Equation

Start by rewriting the given equation. The original equation is \( y - 6 = 0 \). By adding 6 to both sides, you obtain \( y = 6 \).
02

Identify Key Elements

The rewritten equation, \( y = 6 \), is a horizontal line. This is because every point on this line has a y-coordinate of 6, regardless of the x-coordinate.
03

Plot Points Based on Equation

Choose any values for x, since y remains 6. For example, plot points (0, 6), (1, 6), and (-1, 6).
04

Draw the Graph

Using the plotted points, all of which share the y-coordinate of 6, draw a horizontal line on the graph through these points.

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Key Concepts

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

Linear Equations
Linear equations are foundational in algebra and represent straight lines on a graph. A linear equation typically takes the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. These equations show a constant rate of change, meaning the slope is always the same.
Linear equations are easy to work with:
  • The graph of the equation is always a straight line.
  • They can be used to predict values due to their consistent behavior.
  • Linear equations can be in slope-intercept form, standard form, or point-slope form.
In the exercise given, the equation \( y - 6 = 0 \) simplifies to \( y = 6 \), exemplifying a type of linear equation that results in a horizontal line.
Horizontal Line
A horizontal line is one of the simplest forms of a line on a graph. It is characterized by having a constant y-value and a slope of zero.
When an equation is given as \( y = c \), such as \( y = 6 \), it represents:
  • A line that cuts across the y-axis at the point \( c \).
  • Every point on the line shares the same y-coordinate but can have any x-coordinate.
In practical terms, horizontal lines can represent constants in real-world scenarios, such as a flat fee or stable measurement over time without any change.
Plotting Points
Plotting points is a fundamental concept in graphing that provides a visual representation of an equation. To plot points based on an equation:
  • Select values for one variable, usually \( x \), and solve for the other, \( y \).
  • Pair the \( x \) and \( y \) values to form coordinates like \((x, y)\).
  • Place these coordinates on a graph, where the horizontal axis represents \( x \) and the vertical axis represents \( y \).
For the horizontal line \( y = 6 \), you can choose any \( x \) value. This equation implies that the plotted points might be \((0, 6), (1, 6), (-1, 6)\), etc., creating a straight line made entirely of level points.
Quadrants of the Cartesian Plane
The Cartesian plane is divided into four quadrants, making it easier to locate points in a two-dimensional space. These quadrants are organized based on the signs of coordinates:
  • Quadrant I: Both \( x \) and \( y \) are positive.
  • Quadrant II: \( x \) is negative, \( y \) is positive.
  • Quadrant III: Both \( x \) and \( y \) are negative.
  • Quadrant IV: \( x \) is positive, \( y \) is negative.
A line like \( y = 6 \) runs parallel to the x-axis through positive and negative parts of the Cartesian plane, intersecting Quadrants II and I evenly, displaying symmetry over the y-axis. Understanding these quadrants helps in accurately plotting points and interpreting graphs.

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