Exploring Linear Systems: Understanding the Significance of a Single Solution

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In summary, a linear system of equations can have no solution, a single solution, or an infinite number of solutions. The number of solutions depends on the specific equations in the system and can vary. Without knowing more information about the specific system, it is impossible to determine the number of solutions it has.
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nhrock3
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why lenear system has a single solution?
 
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  • #2
In general, a linear system of equations can have no solution, a single solution, or an infinite number of solutions. Do you have a question about a particular linear system?
 
  • #3
it for electrical circuit theory

for thevenin theorem

i have a pi system which has independent sources

and my prof said

"we have a lenear system so the is a single solution"

i can't understand why?
 
  • #4
I don't know what a "pi system" is. In any case, unless you know something about the linear system, you can't tell how many solutions it has.

Here are three very simple examples:

1. No solutions
x + 2y = 3
x + 2y = 4

2. Exactly one solution
x + y = 3
x - y = 1
(solution is x = 2, y = 1)

3. Infinite number of solutions
x + y = 3
2x + 2y = 6
 
  • #5
Note Mark44's options: one solution, no solution, infinite number of solutions.

If you know the problem has a non-zero finite number of solutions then you can say there is a single solution- as opposed to, say, a quadratic equation in which there can be two solutions.
 

FAQ: Exploring Linear Systems: Understanding the Significance of a Single Solution

1. What is a linear system?

A linear system is a set of linear equations that are related to each other. Linear equations are mathematical expressions that involve variables raised to the first power and are graphically represented as straight lines. When these equations are combined, they form a system that can be solved to find the values of the variables that satisfy all of the equations.

2. Why is it important to understand linear systems?

Linear systems are used in many real-world applications, such as in engineering, economics, and physics. Understanding how to solve linear systems allows us to analyze and make predictions about these systems, which can help us make informed decisions.

3. What is the significance of a single solution in a linear system?

A single solution in a linear system means that there is only one set of values for the variables that satisfies all of the equations in the system. This is significant because it tells us that the system has a unique solution and there is no ambiguity in the values of the variables.

4. What does it mean if a linear system has no solution?

If a linear system has no solution, it means that there is no set of values for the variables that satisfies all of the equations in the system. In other words, the lines representing the equations do not intersect, and there is no point that satisfies all of the equations at the same time. This could happen when the equations are parallel or when they represent contradictory statements.

5. How can the number of solutions in a linear system be determined?

The number of solutions in a linear system can be determined by graphing the equations and looking at the intersection of the lines. If the lines intersect at one point, there is a single solution. If the lines are parallel, there is no solution. If the lines overlap, there are infinitely many solutions. Alternatively, the number of solutions can also be determined by solving the system algebraically using techniques such as substitution or elimination.

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