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Matrix and Linear Equations

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Please provide assistance in understanding how to prove whether or not finite mathematical equations are true or false. I have attached the questions that I am experiencing difficulty on. In your solution, please explain how to prove, i.e., your recommendation on the specific formula I should use. My textbook does not provide the step-by-step explanation that I require to gain a thorough understanding of the concepts. Therefore, I request that your explanation depict how to prove that the answer is either true or false.

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1. True. If those two straight lines are not parallel, they will intersect at some point, exactly at one point, which will be the unique solution - a point lying on both the lines.

2. False. It is true only in case when both lines are coincident (identical). Only in coincidence case we have infinitely many points lying on both the lines.
For example, y = x, and y = x+2 are parallel, but they never intersect. We can not find a single point that lies on both the lines.
However, y ...

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This solution provides a detailed explanation as to why each statement will be True or False.

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Matrix Solutions to Systems of Linear Equations

Matrices are the most common and effective way to solve systems of linear equations. However, not all systems of linear equations have unique solutions. Before spending time trying to solve a system, it is important to establish whether it in fact has a unique solution.

For this Discussion Board, provide an example of a matrix that has no solution. Use row operations to show why it has no unique solution. Also, some matrices have more than one solution (in fact, an infinite number of solutions) because the system is undetermined. (In other words, there are not enough constraints to provide a unique solution.) Provide an example of such a matrix, and show, using row operations, why it is underdetermined.

Additional Information for this assignment
 Provide an example of a matrix that has no solution. Use row operations to show why it has no unique solution.
 Also, some matrices have more than one solution (in fact, an infinite number of solutions) because the system is undetermined. (In other words, there are not enough constraints to provide a unique solution.) Provide an example of such a matrix, and show, using row operations, why it is underdetermined.

 Discuss what are consistent and inconsistent systems
 Create your own example of an inconsistent system of equations
 Write these equations
 Provide the augmented matrix for your equations
 Show the row operations. I suggest you work with 2 variables and keep it simple
 State your conclusion
 Provide graphs of your equations

 An inconsistent example is
2x + y = 5
4x+2y = 8

 An consistent example with many solutions is
2x + y = 4
-6x - 3y = -12

 A trick is to create your own equation, then double your first equation to get a second new equation

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