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Chapter 7: Problem 72

When using the addition or substitution method, how can you tell whether a system of linear equations has infinitely many solutions? What is the relationship between the graphs of the two equations?

Short Answer

Expert verified
A system of linear equations has infinitely many solutions when the addition or substitution method results in a true statement like 0=0. The graphs of these equations would be identical, meaning every point is a solution.

Step by step solution

01

Understanding Infinitely Many Solutions

When a system of linear equations has infinitely many solutions, it means the two equations describe the same line. In other words, every solution to one equation is also a solution to the other equation. Hence, there are infinite points satisfying both equations.
02

Criteria for Infinitely Many Solutions

When using the addition or substitution method, if the process of solving the system results in a statement that is always true, such as 0=0 or 3=3, then the system has infinitely many solutions.
03

Graphical Representation:

For a system of two linear equations to have infinite solutions, the graphs of the equations must be the same line. This means the lines represented by the equations coincide and every point on the line is a solution to both equations.

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