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Chapter 1: Problem 85

Graph each equation in the rectangular coordinate system. $$y-1=0$$

Short Answer

Expert verified
The equation represents a horizontal line at y = 1.

Step by step solution

01

Identify the Given Equation

The equation given is: \[ y - 1 = 0 \]This can be rewritten in a more familiar form.
02

Simplify the Given Equation

Simplify the equation by adding 1 to both sides: \[ y - 1 + 1 = 0 + 1 \]which simplifies to: \[ y = 1 \]
03

Interpret the Equation

The equation \( y = 1 \) indicates that for any value of \( x \), the value of \( y \) is always 1. This represents a horizontal line.
04

Graph the Equation

To graph this equation, draw a horizontal line that intersects the y-axis at \( y = 1 \). 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.

Graphing Equations
Graphing equations allows us to visualize the relationships between variables, like x and y, on a coordinate plane. To graph an equation, you need to understand how variables interact and what they represent.
For example, to graph the equation \( y = 1 \), which we derived from \( y - 1 = 0 \), we need to perform a couple of simple steps:
  • First, isolate y to understand its relationship with x. Here, y is always 1, regardless of x.
  • Next, choose a few x-values, but notice y remains 1.
  • Finally, plot these points on the graph and observe the pattern they form.
The more points you plot, the clearer the line or curve representing the equation becomes. In the case of \( y = 1 \), you get a straight horizontal line.
Rectangular Coordinate System
The rectangular coordinate system, or Cartesian plane, consists of a horizontal x-axis and a vertical y-axis. These two axes intersect at the origin (0,0), creating a grid where any point can be described using coordinates (x,y).
Key components include:
  • The Quadrants: The plane is divided into four quadrants: I (top right), II (top left), III (bottom left), and IV (bottom right).
  • Coordinates: Each point on the plane is located by an ordered pair (x, y).
  • Positive and Negative Axes: The right of the y-axis is positive x, left is negative x; above the x-axis is positive y, below is negative y.
Understanding this system is crucial for graphing equations and identifying where different lines and curves intersect. For our horizontal line \( y = 1 \), we see it crosses the y-axis at 1 and stretches infinitely left and right across the x-axis.
Horizontal Lines
Horizontal lines are unique in that they have no slope; the change in y is zero regardless of changes in x. The general form of a horizontal line equation is \( y = c \), where c is a constant.
  • Key properties include:
  • The line is always parallel to the x-axis.
  • Every point on the line has the same y-value.
For example, consider the equation \( y = 1 \). Every point (x, 1) lies on this line, meaning no matter what x is, y is always equal to 1. To graph it: draw a straight line through the points (0,1), (1,1), (-1,1), etc.
Horizontal lines are simple to understand but vital in various contexts, such as defining boundaries or in solving systems of equations.

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Most popular questions from this chapter

Solve each equation. $$\frac{x-2}{x-3}=\frac{x-3}{x-4}$$

$$\text { Solve } \frac{x}{2}+\frac{1}{3}=\frac{x}{9}+\frac{1}{6}$$.

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