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. 2x - y = 6 4x - 2y = 0

Accepted Answer

Hey, everyone here we are asked to solve the system of equations using the Gauss Jordan method and provide the solution with Y arbitrary for systems in two variables that have infinitely many solutions. Here we are given two equations. The first equation is 17 X minus Y is equal to 11. And the second equation is 34 X minus two Y is equal to zero. Here we have four answer choice options. Answer choice A X is equal to zero and Y is equal to 22. Answer B X is equal to 22 Y is equal to zero. Answer C X is equal to 17 and Y is equal to 22 answer D no solution. So here, the first step in applying the Gauss Jordan method is to first set up our system of equations inside an augmented matrix. So recalling matrices, we first have two large brackets along with a vertical line which represents our equal sign. And now to the right of the vertical line, we have one column which contains the values of the constants from both equations. And to the left of the vertical line, we have two columns one for each variable. And so here we will have the first column B, the X coefficients, which means the second column will be the Y coefficients. So now filling in our matrix here, our first row, we need to look at our first given equation to obtain the values. So we see here that first, our X coefficient is 17. So we place 17 in the first row and first column. Now moving to the next column, we have our Y coefficient which according to our first equation is negative one. And now moving to the right of the vertical line, we have the value of 11 for our constant. Now moving on to filling in our second row, we repeat the same process. But now look at our second given equation and from our second equation, we obtain the values 34 negative two and zero. So now recalling the gosh Jordan method, we need to recall that we need to get our first row and first column value to become one. And so in this case, we need to make 17, take the value of one. Well, that means we just need to take row one and replace it with row one divided by 17. So performing this operation, we need to rewrite our matrix as this new form. And so again, for row one, we are taking each value of row one and dividing it by 17. This gives us new values of one negative one. Divided by 17 and 11, divided by 17. And now row two takes the same values. It remains unchanged as 34 negative two and zero. So now the next step here is to move down the first column to the next value which is 34 we need to make this become zero. So to do this, we take row two and replace it with the operation where we take 34 times row one and then we can subtract row two. So now doing this, we once again rewrite our matrix here we have row one remain unchanged. So it still has the values of one negative one divided by 17 and 11 divided by 17. Now row two performing 34 times row one minus row two, we get the new values of 00 and 22. So now taking a closer look at our current second row, we get the equation zero X plus zero Y is equal to 22. Well, this is the same as saying that zero is equal to 22. And we know that this statement is false, that zero is not the same as 22. And since this statement is false, we can conclude that there is no solution. So again, because our second row produces a false statement, we can conclude that there is no solution to our given system of equations. So here we are left with answer choice D where again, our system of equations has no solution. 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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