 In this video, we will provide the solution to question number nine for the practice exam number one for math 2270. In this question, we are given a statement, a system with more unknowns than equations has at least one solution. And so we're supposed to determine whether this thing is true or false. So we're considering a system of equations with more variables than equations. Is it consistent? If it has at least one solution, that means it would be consistent. Now, if you have more unknowns, the variables correspond to the columns, then equations which correspond to the rows, we're thinking of an augmented matrix which has more columns than rows. And so this is going to be a false statement. Consider the following counter example. Let's take a say a two by three coefficient matrix. It could look something like one, two, three, and then augment that with four. And then you're gonna get one, two, three, augment that with five. Notice that the coefficients in the two equations are identical, but the right hand sides are different. And so if we were to row reduce this matrix, notice this would row reduce to be one, two, three, four. And then we get zero, zero, zero, one, right? If we took row two minus row one. And so this would be an example of an inconsistent system. In fact, we could potentially start with something like this, but to really kind of fill out here, consider this thing right here. And so notice what this counter example presents to us. The above augmented matrix corresponds to a linear system with three variables, three unknowns and two equations, but it is inconsistent, inconsistent. I hope I spelled these words right. And so this would actually provide us a counter example to show that a statement is false. You provide a counter example and give some evidence on why this counter example in fact does show that the statement is false. One thing you should be aware about with true or false statements. If a statement is true, that means it's always true. That means for every possible assignment it's true. If a statement is sometimes true or sometimes false, then actually means it's false. Because if counter examples exist, that means the statement is false. One counter example is enough to show that a statement is false. To show that a statement is true, you'd have to then provide an argument why it's always true.