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 = 5 x - y = -1

Accepted Answer

Hey, everyone. Here we are asked to solve the system of equations using the Gastro method and provide the solution with Y arbitrary for systems in two variables that have infinitely many solutions. So here we are given a system with two equations where the first equation is plus Y is equal to 25. And the second equation is two X minus Y is equal to 11. Here we have four answer choice options, answer choice A X is equal to 11 and Y is equal to 14. Answer B X is equal to 14 and Y is equal to 11. Answer C X is equal to 12 and Y is equal to 13 and answer D X is equal to 13 and Y is equal to 12. So here our step before we can start using the Gastro method is to set up our system of equations inside an augmented matrix. So recalling matrices, we first have two large brackets along with a vertical line that represents our equal sign. And to the right of this vertical line, we will have one column including the values of the constants from our equations. And then to the left of the vertical line, we will have two columns, one for each variable where the leftmost column will be the X coefficients. And then the second column will be the Y coefficients. So now filling in our matrix beginning with the first row, we need to look at our first equation for the values. And we see that the X coefficient is one, the Y coefficient is also one and the value of our constant is 25 moving on to the second row. We now look at our second equation and we get the values two negative one and 11. So now that we have set up our augmented matrix, we can begin applying the gas Jordan method where we first have to change the value which lies in the first row and first column so that it is one. Well, here we see that the value is already one. So we can just move down to the next value you in the first column. And we see this value is two. Well, we want this two to become zero. So we need to perform a row operation where we take row two and we replace it with negative two times row one plus row two. And now with this row operation, we will have new values for our second row. So that means we need to rewrite our matrix with these new values. So first, we have our first row which remains unchanged with the values 11 and 25. But now our second row has new values which are zero, negative three and negative 39. So now that we have finished manipulating our first column, we can move on to our second column where we first want to focus on the last value which is negative three and we want this negative three to become one. So we simply take row two and replace it with row two divided by negative three. And now here we are taking each value in row two and dividing it by negative three. So that means once again, we need to rewrite our matrix with our new values. And so once again, the first row remains the same with the values 11 and 25. But our second row has new values which are 01 and 13. So now we can move back up to the first value in the second row, which is one and we need this one to become zero. So we simply take row one and replace it with row one minus row two. And now with this final row operation, we rewrite our matrix to its simplified form here. So our first row now obtains new values which are 10 and 12 and our second row is still 01 and 13. So now we have fully manipulated our original matrix according to the gas Jordan method. So we just need to recall that the first column contains the X coefficients, the second column contains the Y coefficients. The vertical line represents the equal sign and the column to the right of the vertical line holds the constant values. So now we also need to recall that each row is in a matrix represents one equation. So here we have two rows which means we have two equations. And from the first row, we get the equation one X plus zero Y is equal to 12. And from the row, we get the equation zero XX plus one Y is equal to 13. So now our last step is to just simplify our equations here. And so our first equation becomes X is equal to and our second equation becomes Y is equal to 13. So using the gosh method, we have found our values for X and Y and so we are left here with answer choice C where X is equal to 12 and Y is equal to 13. 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

Augmented Matrix

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