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In Exercises $66-69$, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If a linear system has two distinct points that are solutions, then the graphs of the system's equations have equal slopes and equal $y$ -intercepts.

Name: Problem 68, Chapter 4: Introductory and Intermediate Algebra for College Students 4th
Uploaded: 2020-06-09T17:57:00-08:00
Duration: 9 s
Channel: John Irizar
Description: In Exercises 66-69, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If a linear system has two distinct points that are solutions, then the graphs of the system's equations have equal slopes and equal y -intercepts.

   In Exercises $66-69$, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If a linear system has two distinct points that are solutions, then the graphs of the system's equations have equal slopes and equal $y$ -intercepts.

Introductory and Intermediate Algebra for College Students 4th

Robert Blitzer 4th Edition

Chapter 4, Problem 68

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Solved by Expert John Irizar

06/09/2020

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This means that there are two points that satisfy both equations in the system. Show more…

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In Exercises $66-69$, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If a linear system has two distinct points that are solutions, then the graphs of the system's equations have equal slopes and equal $y$ -intercepts.

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Key Concepts

Graphical Attributes of Lines

The slope and y-intercept are key characteristics that define the graphical representation of a linear equation. For two lines to be identical, as is the case in a dependent system, they must have the same slope and the same y-intercept, ensuring their graphs coincide completely.

Linear Systems

A linear system consists of two or more linear equations considered together, where the solutions to the system are the set of points that satisfy all equations simultaneously. In the context of two equations in two variables, if the system has more than one common solution, it implies that the equations represent the same line.

Uniqueness of a Line Determined by Two Points

In analytic geometry, two distinct points uniquely determine a straight line. This means that if two distinct points are common solutions to two linear equations, then both equations must represent the same line because only one line can pass through those two distinct points.

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