True or False: A Linear System Must Have a Unique Solution

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In summary, a linear system is a set of equations represented by straight lines on a graph that can be solved using algebraic methods. A unique solution means there is only one set of values that satisfies all equations, and can be determined by intersecting lines on a graph. If a linear system does not have a unique solution, it can have no solutions or an infinite number of solutions. To determine if a linear system has a unique solution, the number of equations must be equal to the number of variables and they must be independent. Having a unique solution is important because it allows for accurate predictions and real-world problem solving, and ensures consistency in the system.
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Valour549
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So I'm reading chapter 1.1 of Elementary Linear Algebra by Anton and there's this true-false question:

"A linear system whose equations are all homogenous must have a unique solution."

Taking this question simply as it is I think the answer should be false, because while there's always the trivial solution, it's also possible for there to be infinite solutions. Yet the book gives the answer as True? I think it's pretty confusing and doesn't give the student confidence when the very first question has the wrong answer.
 
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  • #2
The book is wrong.
 

What is a linear system?

A linear system is a set of equations that can be represented by a straight line on a graph. It involves variables and coefficients that can be solved using algebraic methods.

What is a unique solution?

A unique solution in a linear system means that there is only one set of values for the variables that satisfies all of the equations. In other words, it is the point at which all of the lines intersect on a graph.

What happens if a linear system does not have a unique solution?

If a linear system does not have a unique solution, it means that there are either no solutions or an infinite number of solutions. This can happen if the equations are parallel or if they are equivalent, meaning they have the same slope and y-intercept.

How can you determine if a linear system has a unique solution?

A linear system has a unique solution if the number of equations is equal to the number of variables and the equations are independent, meaning they are not multiples of each other. This can be determined by using methods such as elimination or substitution.

Why is it important for a linear system to have a unique solution?

A unique solution in a linear system allows us to find specific values for the variables and make accurate predictions or solve real-world problems. It also ensures that the system is consistent and that there is a single solution that satisfies all of the equations.

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