Form 8655 Reduced 2024

I have here three equations of four unknowns. You can already guess or you already know that if you have more unknowns than equations, you're probably not constraining it enough. So, you're actually going to have an infinite number of solutions. But those infinite number of solutions could still be constrained within, well, let's say that we're in four dimensions. In this case, we have four variables. Maybe we're constrained into a plane in four dimensions. Or, if we're in three dimensions, maybe we're constrained to a line. A line is an infinite number of solutions, but it's kind of a more constrained set. So, let's solve this set of linear equations. We've done this by elimination in the past, but what I want to do is I want to introduce the idea of matrices. Matrices are really just arrays of numbers that are shorthand for this system of equations. So, let me create a matrix here. I could just create a coefficient matrix where the coefficient matrix would just be the coefficients on the left-hand side of these linear equations. The coefficient there is one, coefficient there is one, coefficient there is two, two, four, two, two, four, one, two, zero, one, two, there's no coefficient on the x3 term here because there's no x3 term. Now, if I just did this right there, that would be the coefficient matrix for this system of equations, but what I want to do is I want to augment it. I want to augment it with what these equations need to be equal to. So, let me uncomment it, and what I'm going to do is I'll just draw a little line here and write the 7, the 12, and the 4. And I think you can see that whatever this right here...