### Video Transcript

Fill in the blank. For the system of equations defined
by two equations of three variables, the size of the augmented matrix is blank.

A general system of linear
equations in the variables 𝑥 one, 𝑥 two up to 𝑥 𝑛 and coefficients 𝑎 𝑖𝑗 looks
like this. Then, another way of presenting
this information is in what we call an augmented matrix, and that looks like
this. This augmented matrix represents
the same information just in a different way. We can see that the coefficients of
the system of linear equations appear on the left of the augmented matrix. So the number of entries in this
augmented matrix varies depending on the number of equations and the number of
variables that we have.

So, for a system of equations
defined by two equations of three variables, a system of two equations of three
variables will look like this. We can see that we have two
equations and we have three variables. That’s 𝑥 one, 𝑥 two, and 𝑥
three. We can see straightaway that
there’s going to be six coefficients here. So our augmented matrix will look
like this. We will have our six coefficients
on the left, and we’ll have our two constants on the right. That’s 𝑏 one and 𝑏 two. So that is our augmented matrix for
a system of two equations and three variables. And we can see that this is a
two-by-four matrix because it has two rows and four columns.

One mistake that you can make with
this kind of question is only considering the order of the coefficient matrix. But remember, the augmented matrix
includes the constants too, making it a two-by-four matrix. So remember, the augmented matrix
will always have the same number of rows as the number of equations and one more
column than the number of variables.