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Matrix and Linear Equations

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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Solution Preview

See the attachment.

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 ...

Solution Summary

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