In Exercises 1–8, write the augmented matrix for each system of l...

Question

In Exercises 1–8, write the augmented matrix for each system of linear equations.

Accepted Answer

hey everyone here we are asked to determine the augmented matrix that corresponds to the given system of equations. So our first given equation is three, W plus two, X minus four, Y is equal to five. Our second equation is two, W minus eight, X plus nine, Y is equal to zero. Our third equation is given as 10 x minus three, Y is equal to one. And then the fourth and final equation is w minus two X minus six, Y plus Z is equal to 11. And so our first step here is to show each coefficient for each variable. So what I mean is beginning with this first equation here we see that we have three variables but our total system has a total of four different variables. So here we have three W plus two X minus four Y. But we are missing our Z variable. So I'm going to add zero Z. So we see the coefficient for Z. And then we have is equal to five. And now we can move on to our second equation here we have two w minus eight X plus nine Y. And we are also missing our z variable. So let's add zero Z. And then we have is equal to zero. And now for our third equation we have two variables, we have 10 X. And we have minus three Y. But we need to add the plus zero Z. And we also are missing the W variable. So we need a plus zero W. And then this equation is equal to one and now for our final equation here we have each of our variables but we see that W and Z do not have a a coefficient. This is because when a variable is on its own, the one is implied. But for clarity I'm going to write the coefficients for both of these variables. So we really have one, w minus two, X minus six, Y plus one Z equal to 11. And so now we clearly have shown each coefficient for each variable. So we can go ahead and write these equations into an augmented matrix. So if we recall with an augmented matrix, we have this vertical line here to represent the equal sign to the right hand side. This column here will have all constants and then to the left hand side, each of these terms will be the coefficients from each given variable. So in this case we have going from left to right, we have W, X, y and Z coefficients for each column. Now we can begin filling in our matrix beginning with row one. We look at our first equation and we have the coefficients here. So we have three W. Then we have 24 X, then negative 44 Y and 04, Z is equal to five, which we place on the right hand side. Now we can move on to our second row where we look at our second equation and we have the values to negative 89 and zero is equal to zero. Thirdly we have the third equation with the values 10 negative 30 is equal to one. And lastly we have the values from our fourth equation, which give us the fourth row. One negative two negative 61 is equal to 11. And with this we are left with answer choice D. And we have our finished augmented matrix. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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In Exercises 1–8, write the augmented matrix for each system of linear equations.