Video Transcript
Find the augmented matrix for the following system of equations: two 𝑥 plus nine 𝑦 plus two equals zero; three 𝑦 minus four 𝑥 minus six equals zero.
A system of linear equations is where we have one or more linear equations involving a set of variables. In this case, we have two linear equations involving the variables 𝑥 and 𝑦. We refer to the numbers that come before the variables as coefficients. There is an alternative way to present a system of linear equations, and that is in an augmented matrix. In general, for a system of linear equations in the variables 𝑥 one, 𝑥 two to 𝑥 𝑛 and coefficients 𝑎 𝑖𝑗, we can write the system of linear equations as an augmented matrix, and it looks like this. The matrix is split into two parts. And on the left part, we have what we call the coefficient matrix. That is, as the name suggests, the matrix of all the coefficients from this system of linear equations.
Notice how the coefficients are not just in any order but the same order as they’re written above in the linear equations. The coefficients for a certain variable stay within the column. And on the right of the augmented matrix, we have these values. So a good place to start for our question is rewriting this system of linear equations in this way. So the first thing I’ve done is just rearranged the second equation so that the 𝑥’s come first followed by the 𝑦’s, just in the same way as it’s written in the first equation. And this just keeps the coefficients of 𝑥 lined up and the coefficients of 𝑦 lined up. This makes it much easier to put it in the augmented matrix form.
The next thing I’m going to do is rearrange both of the equations so that these values, the values that aren’t attached to variables, appear on the right-hand side of the equals. So my system of linear equations is two 𝑥 add nine 𝑦 equals negative two and negative four 𝑥 add three 𝑦 equals six. So now I’ve written it in this way, it’s going to be fairly straightforward to put this into the augmented matrix form. Let’s begin by setting up the matrix. Remember that on the left, I’m going to have the coefficient matrix, so the matrix of the coefficients. So on the top left, I’m going to have two. And on the bottom left, I’m going to have negative four. So this column represents the coefficients of 𝑥.
The next column will consist of the coefficients of 𝑦, that is, nine and three. And then the right-hand part of this augmented matrix will consist of the values negative two and six. And that gives us our solution. This is the augmented matrix for this system of linear equations. When answering a problem like this, it’s really important to check that you have the coefficients of the same variable lined up before you put it into the matrix.
