 In this video, we're going to provide the solution for question number eight on the practice exam number one from math 2270. You are asked to express the following linear system given right here as a vector equation and as a matrix equation. So we want to identify the column vectors in this thing. Make sure they're all lined up and they already are x1, x2, x3. Any constants are on the right-hand side. So we're good to go. So in terms of the vector equation, we're going to have an x1 times the first column of coefficients, which is three, zero, one and four. Just recording these numbers right here, the coefficients. Then we're going to add to that x2 times the next column of coefficients, which is zero, two, negative two and negative five. So we're recording these coefficients right here. Make sure you grab the negatives when you have them. And if you ever see a blank, that's because it's a zero. Then next, we're going to get x3. We get a vector for each of the variables. We're going to get negative three, negative one, two and zero for the third column. And then the last, the column on the vector on the right-hand side of this equation will just be the constants you see right here. So this would be four, five, seven and one. This then gives us the vector equation. The corresponding matrix equation, it should have the form ax equals b, where the matrix a is going to be the coefficient matrix whose columns are actually these columns right here. We can also record them directly from this. We're going to get three, zero, one, four first column, zero, two, negative two, negative five second column, negative three, negative one, two and zero. This gives us the matrix a, the coefficient matrix. Then we times this by the vector x, which is a variable vector. It'll be x1, x2, x3. This will just be the variables in play. And then this is equal to the right-hand side of those equations, the vector b, which is four, five, seven and one. So you can see that the right-hand sides of the matrix equation and the vector equations will be identical. And they correspond to the right-hand sides of these things right here. Now be aware that the matrix equation is not the same thing as the augmented matrix. I'm not asking for that. The matrix equation means this thing right here. You have this four by three matrix times by this vector from R3, and it's equal to a vector from R4. That's what the matrix equation is going to look like. And this one, there's really not a lot of work to show. It's just we want to write the vector equation and the matrix equation. But be aware, this could ask for the matrix equation and the augmented matrix, the vector equation, the augmented matrix. Or you might actually be given the matrix equation. You're asked for the linear system of the matrix equation or something like that. Be prepared for those variants on this question.