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Chapter 3: Problem 3

Determine whether each equation is a linear equation in two variables. See Example 1. $$ x=y $$

Short Answer

Expert verified
The equation \( x = y \) is a linear equation in two variables.

Step by step solution

01

Identify the Standard Form of a Linear Equation in Two Variables

The standard form of a linear equation in two variables is often represented as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants and \( A \) and \( B \) are not both zero.
02

Rewrite the Given Equation in Standard Form

The given equation is \( x = y \). To analyze it, let's rewrite it in the standard form of a linear equation: \( x - y = 0 \). In this form, we can see \( A = 1 \), \( B = -1 \), and \( C = 0 \).
03

Determine if the Constants Meet the Conditions

In the standard form \( Ax + By = C \), the equation is linear if \( A \) and \( B \) are not both zero. Here, \( A = 1 \), \( B = -1 \), and \( C = 0 \). Since \( A \) and \( B \) are not both zero, the equation is linear.

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Key Concepts

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

Standard Form
In linear algebra, the standard form of a linear equation helps to keep equations organized and easy to read. A linear equation in two variables is typically expressed as \( Ax + By = C \). This form is highly beneficial for quickly identifying the coefficients and the constant term of an equation.
Standard form has several advantages:
  • It provides a uniform way to handle equations, especially when dealing with systems of linear equations.
  • The coefficients \( A \), \( B \), and the constant \( C \) are readily apparent, making it easier to analyze the equation.
In the context of graphing, the standard form can help determine the intercepts and the slope of the line. The standard form is versatile, fitting well with multiple mathematical techniques and allowing easier solving and understanding of complex problems.
Two Variables
Linear equations in two variables involve two distinct variables, often represented as \( x \) and \( y \). The presence of two variables means you will visualize their relationship as a straight line on a coordinate plane.
Understanding these key points will help:
  • Each solution to the equation can be viewed as a point (or ordered pair) \((x, y)\) on the coordinate plane.
  • The line representing the equation consists of all points that satisfy it.
  • The equation \( x = y \) can be rewritten in the standard form \( x - y = 0 \), demonstrating how \( A = 1 \), \( B = -1 \), and \( C = 0 \).
The equation defines a linear relationship between the variables, implying a constant rate of change, or a straight line when graphed. This representation makes it easier to understand and solve problems involving linear relationships.
Constants in Equations
Constants in an equation like \( Ax + By = C \) play a crucial role. They define the specific characteristics of the linear equation. In our example of \( x - y = 0 \), the constants \( A = 1 \), \( B = -1 \), and \( C = 0 \) provide the structure of the equation.
Why constants are important:
  • Constants \( A \) and \( B \) are not both zero; otherwise, it would not represent a linear equation.
  • The constant \( C \) indicates the position where the line crosses the axis (when rewritten in an intercept form).
  • All the constant values together affect the slope and intercept of the line.
By knowing the values of these constants, you can predict and understand the behavior of the linear equation's graph.

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Most popular questions from this chapter

One of Japan's superconducting "bullet" trains is researched and tested at the Yamanashi Maglev Test Line near Otsuki City. The steepest section of the track has a rise of 2580 meters for a horizontal distance of 6450 meters. What is the grade of this section of track? (Source: Japan Railways Central Co.)

Find the slope of the line that is (a) parallel and (b) perpendicular to the line through each pair of points. \((6,-1)\) and \((-4,-10)\)

For Exercises 87 through \(91,\) fill in each blank with "0, "positive," or "negative." For Exercises 92 and \(93,\) fill in each blank with "x" or "y." (__,__) quadrant I

For Exercises 87 through \(91,\) fill in each blank with "0, "positive," or "negative." For Exercises 92 and \(93,\) fill in each blank with "x" or "y." (__,__) quadrant IV

Write the equations of three lines parallel to \(10 x-5 y=-7\) (See the third Concept Check.)
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