Write the augmented matrix for each system of linear equations...

Question

Write the augmented matrix for each system of linear equations.

$\begin{cases}x - y + z = 8 \y - 12z = -15 \z = 1\end{cases}$

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 just X is equal to five. Our second equation is x minus four, Y is equal to one. And our third and final equation is two, X plus six, Y plus seven, Z is equal to negative 12. Now our first step in solving this problem is to write out the coefficient of each variable. So beginning with our first equation here, we just have X is equal to five. Well, since X is all on its own, we know that when a variable is alone, the one coefficient is implied. So we can just add the one in front of X. We then also notice that we only have X and we're missing varia Y and Z. Well this tells us that in this equation the coefficient of both of these variables are just zero. So we can just add plus zero Y and plus zero Z. Now we are showing the coefficients of each variable for this first equation. So we can go ahead and move on to the second equation where we have x minus four, Y again is equal to one. Well here, just like the first problem or sorry, first equation we have X on its own. So we can just add the one in front of it. We then are also missing the Z variable, which tells us that Z here has a coefficient of zero. So we can just add plus zero Z. And now we can move on to our final equation which actually shows each variable each coefficient for each variable. So we can just rewrite this equation as two X plus six, Y plus seven Z is equal to negative 12. And with this since we are now clearly showing each coefficient, we can go ahead and write into our augmented matrix. So if we recall with an augmented matrix we have this vertical line to represent the equal sign to the right hand side of this line, we have the constants from each equation and to the left hand side we have the coefficients for each given variable. So starting from left to right we have x coefficients as our first column. Then we have the y coefficients and lastly the coefficients. So now with this information we can go ahead and start filling in our matrix for row one. We look at the first equation and we see our coefficient. So we have one for X, then 04, Y and 04 Z. And our constant here is five. So we place it on the right hand side. Next we can move on to our second equation for the second row and we see the values one negative four and zero for the left hand side. And on the right hand side we have is equal to one. So we'll just place the value one and now we can move on to our third equation for our final row and we have the values to 67 and negative 12 on the right hand side. And with this we have completed our augmented matrix, which leaves us with answer choice. C. Thanks for watching. I hope you found this video helpful, and I'll see you again next time.

Write the augmented matrix for each system of linear equations.
$\begin{cases}x - y + z = 8 \\y - 12z = -15 \\z = 1\end{cases}$

Key Concepts

System of Linear Equations

Augmented Matrix

Matrix Notation and Construction

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