Use the Gauss-Jordan method to solve each system of equations. Fo...

Question

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Examples 1-4. x + y - 5z = -18 3x - 3y + z = 6 x + 3y - 2z = -13

Accepted Answer

Hey, everyone. Here we are asked to solve the system of equations using the gastro method. And if the system has infinitely many solutions express these solutions set with Z being arbitrary. So here we are given a system with three equations where the first equation is X plus Y minus two, Z is equal to negative 22. The second equation is five, X minus five Y plus Z is equal to negative 41. And the third equation is two X plus Y minus three C is equal to negative 38. Here we have four answer choice options, answer choice A X is equal to negative three, Y is equal to seven and Z is equal to nine. Answer B X is equal to negative seven, Y is equal to nine and Z is equal to three. Answer C X is equal to negative nine, Y is equal to seven and Z is equal to three and answer D X is equal to negative seven, Y is equal to three and Z is equal to nine. Now, before we can begin using the gastro method, we first need to set up our system of equations inside an augmented matrix. So recall matrices, we first have two large brackets along with a vertical line that represents our equal sign from our equations. And to the right of the vertical line, we will have one column which holds the values of the constants. And then to the left of the vertical line, we have three columns. Since we have three variables. And starting in alphabetical order, the leftmost column will hold the X coefficients. Then the second column holds the Y coefficients and then the third column holds the Z coefficients. So now filling in our matrix, starting with our first row, we look at our first given equation for the values. And here we see the X coefficient is one, the Y coefficient is also one, the Z coefficient is negative two and the value of our constant is negative 22. Moving on to our second row. We now look at our second equation and we get the values five negative 51 and negative 41. And now from our third equation, we can fill in our third row with the values which are 21, negative three and negative 38. So now that we have finished setting up our augmented matrix, we need to begin performing row operations according to the Gosden method. So the first step in this method is to change the value which lies in the first row and first column so that it is one. And we see that this value is already one. So we just need to move down to the second row in the first column. And here the value is five. Well, we, we want this five as well as the value below it, which is two, we want both of these values to become zero. So beginning with our second row, we need to take row two and replace it with row two minus five times row one. And so this row operation will cause our five in the first column to become zero. And now, since we also want the two below it to become zero, we perform an additional row operation where we take row three and then we replace it with row three minus two times row one. So now with these two row operations, we need to rewrite our matrix since we now have new values. And so here our first row remains unchanged. So its values are still 11, negative two and negative 22. And now our second row has new values which are zero, negative 10, 11 and 69. And then the third row also has new values which are zero, negative 11 and six. So now we have finished manipulating our first column according to the gas shorten method. So now we can move on to the second column where we first want to focus on the second row, which has the value negative 10. And we want this negative 10 to become one So all we need to do is take our second row and replace it with row two divided by negative 10. So now we are just dividing each value in row two by negative 10. However, this gives us new values. So that means we again need to rewrite our matrix. So here the first row is still the same with the values 11, negative two and negative 22. Now our second row has new values which are 01, negative 11 divided by 10 and negative divided by 10. And then our third row is still the same as the previous matrix. So it has the values zero, negative 11 and six. Now staying in our second column here, we want to now focus on the first value which is one and the last value which is negative one. And we want both of these values to become zero. So starting in our first row, we will take row one and replace it with the operation where we have just row one minus row two. And now in our last row here, again, we're manipulating this negative one. So we need to take our entire third row and we replace it with row three plus row two. So now we have two additional row operations. So again, we need to rewrite our matrix with our new values. So here our first row has new values which are 10, negative nine divided by 10. And negative 151 divided by 10. Our second row is unchanged from the previous matrix. So its values are still 01, negative 11 divided by 10 and negative 69 divided by 10. And now our third row has new values and these values are 00, negative one divided by 10 and negative nine divided by 10. So now we have finished with our second column. So we want to move on to our third column and we first need to focus on the last value in our third column, which is negative one divided by 10. And we need this value to become one. So all we have to do is take row three and replace it with negative 10 times row three. And with this additional row operation, we once again rewrite our matrix with the new values. So our first row is unchanged. So we have the values 10, negative nine divided by 10 and negative divided by 10. Our second row remains as 01, negative 11 divided by 10 and negative 69 divided by 10. And now our third row has new values which are 001 and six or sorry one and nine. And so now staying in our third column here, looking at the first two values which are negative nine divided by 10 and we have also negative 11 divided by 10. Well, we want both of these values to become zero. So we have two final row operations starting in our first row, we take row one and we replace it with row one plus nine divided by 10 times row three. And now in our second row, we take our second row and we replace it with row two plus 11 divided by 10 times row three. And so with these final row operations, we need to rewrite our matrix. Now, with our final simplified form according to the gas Jordan method where the first row now has the values zero and negative seven. Our second row now has the values 010 and three. And our third row remains with the values 001 and nine. So now that we have finished applying the Gauss Jorden method, we just need to recall that the first column contains the X coefficients. The second column contains the Y coefficients. The third column contains the Z coefficients. The vertical line represents the equal sign and the column to the right of the vertical line holds the values of the constants. And next, we also need to recall that each row in a matrix represents one equation. So here, since we have three rows, we have three different equations. And so from our first row, we get the equation one X plus zero, Y plus zero, Z is equal to negative seven. From our second row, we get the equation zero X plus one Y plus zero, Z is equal to three. And then from our third row, we get the equation zero X plus zero, Y plus one C is equal to nine. And so now our last step is to just simplify each of these equations. So from our first equation, we have just X is equal to negative seven. Then from our second equation, we have Y is equal to three. From our third equation, we have Z is equal to nine. So here from applying the gosh Joan method, we have found values for X Y and Z and so we are left here with answer choice D where again, X is equal to negative seven, Y is equal to three and Z is equal to nine. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Key Concepts

Gauss-Jordan Elimination

Infinitely Many Solutions

Parametric Representation

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