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. 6x - 3y - 4 = 0 3x + 6y - 7= 0

Accepted Answer

Hey, everyone. Here we are asked to solve the system of equations using the cost shorted method and provide the solution with Y arbitrary for systems in two variables that have infinitely many solutions. Here we are given a system of two equations where the first equation is four X minus two, Y plus one is equal to zero. And the second equation is two X plus four, Y minus three is equal to zero. Here we have four answer choice options. Answer choice A X is equal to 11 divided by 10 and Y is equal to seven divided by 10. Answer B X is equal to one divided by 10 and Y is equal to seven divided by 10. Answer C X is equal to seven divided by 10 and Y is equal to one divided by 10 and answer D X is equal to one divided by 10 and Y is equal to divided by 10. So here to utilize the Gauss Jorden method, we first need 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 the vertical line, we will have one column for our constant values to the left of the vertical line. We will have two columns, one for each variable where the leftmost column starting in alphabetical order will be the X coefficients. And that means the second column will be the Y coefficients. So now looking at our two given equations, we see that the constants are on the left side of the equal sign. Well, we first want to just rearrange these equations so that we have the constant on the right side. And so doing this, our first equation becomes four, X minus two Y is equal to negative one. And the second equation becomes two, X plus four Y is equal to three. So now that we have rearranged our given equations, we can go ahead and start filling in our matrix. So starting with the first row, we look towards our first rewritten equation to find our values. And we see that the coefficient of X is four, then the coefficient of Y is negative two. And lastly the value of the constant is negative one. Now moving on to the second row, we now look at our second rewritten rewritten equation and we have the values 24 and three. So now that we have filled in our matrix, we can begin applying the Gauss Joan method. And so the first step is to look at the value which is in the first row and first column. In this case, this value is four and we want this four to become one. So that means we need to take row one and then we replace it with the operation where we have row one divided by four. And this works because we know four divided by four gives us one, which is what we are, what our goal is. So now applying this row operation, we need to rewrite our matrix with our new values. So row one obtains new values which are one, negative one half and negative 1/4. And now row two remains unchanged. So it keeps its original values which are 24 and three. So now the next step is to move down our first column to the last value here, which is two. And so we take this two and we need it to become zero. So that means we need to manipulate our second row. So we take row two and we replace it with the operation where we have two times row one minus row two. And now with this operation, once again, we rewrite our matrix. So our first row now remains unchanged. And so it keeps its previous values which are one, negative, one half and negative 1/4. And now the second row obtains new values which are zero, negative five and negative seven halves. So now that we have finished manipulating our first column according to the gosh, Joan method, we need to move on to our second column. And so in our second column, we need to take a look at the last value which is negative five and we need this negative five to become one. So that means we just take row two and we replace it with row two divided by negative five. So now once again, we need to rewrite our matrix. And so now with our second row, we divide each value by negative five. So it obtains new values which are and seven divided by 10. However, our first row remains the same. So it still has the values one negative one half and negative 1/4. So now staying in our second column, we move back up to the first value which is negative one half and we need this negative one half to become zero. So we will take row one and we will replace it with one half times row two plus row one. And so with this final row operation, we once again rewrite our matrix with the new values for row one. And so row one has the new values 10 and one divided by 10. And now the second row is still the same. So it has the values 01 and seven divided by 10. So now that we have fully manipulated our matrix according to the Gas Jordan method, we just need to recall that our first column has the X coefficients. The second column is the Y coefficient, the vertical line represents an equal sign and the third column or the column to the right of the vertical line has the values of the constant. And so now we just need to note that for each row in a matrix, we get one equation. And so here we have two rows, so we will get two individual equations. And so looking at our first row, we get the equation one X plus zero Y is equal to one divided by 10. And from our second row, we get our second equation which is zero X plus one, Y is equal to seven divided by 10. So now we just need to simplify both of our equations. So the first equation simplifies two X is equal to one divided by 10. And now the second equation simplifies two Y is equal to seven divided by 10. So now utilizing the gastro method, we have found our values for X and Y. And so we are left here with answer choice B where again, X is equal to one divided by 10 and Y is equal to seven divided by 10. 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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