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. 3/8x - 1/2y = 7/8 -6x + 8y = -14

Accepted Answer

Hey, everyone. Here we are asked to solve the system of equations using the gas Jorden method and provide the solution with Y arbitrary for systems and two variables that have infinitely many solutions. Here we have two given equations. The first equation is 7/15 X minus one third Y is equal to 3/5. And the second equation is negative 21 X plus 15 Y is equal to negative 27. Here we have four answer choice options, answers A through D. Each of them being a solution set containing one solution as the variable Y itself and the other solution is some expression containing the variable Y. So to begin solving this problem using the gas Jordan method, we first need to set up our system of equations in an augmented matrix. So recalling an augmented matrix, we first have these two large brackets along with a vertical line which represents our equal sign from the equations. And now to the right of the vertical line, we place the values of the constants from the equations. And to the left of the vertical line, we will have two columns. The first column will be the X coefficient. And the second column will be the Y coefficients. Now beginning with filling in the values for our first row, we obtain the values from the first given equation. So first, we can start with the coefficient of X which is 7 15. Now for the coefficient of Y, we have negative one third. And lastly for the value of the con to the right of the vertical line, we have 3/5. Now moving on to the second row, we obtain our values the same way. But now using the second given equation. So our second row has the values negative 15 and negative 27. So now recalling the gos Jordan method, we want to make it such that our value in the first column and second row, which in this case is negative 21 we want this value to become zero. So before we can get this value to become zero, we want to manipulate our matrix such that both of the X coefficients are the same same. So we can start with manipulating our first row. So we will take row one and replace it with 15 times row one. Now moving to our second row again, we want our X terms to be the same. So we will take row two and then replace it with row two divided by negative three. Now performing these row operations, we can rewrite our matrix as a different form. After using or applying these operations. So rewriting our matrix beginning with the first row, again, we take the first row and multiply each term by 15. And with this, we get our new values which are seven negative five and nine. Now moving on to the second row. Now here for the second row, we divide each term by negative three. So we get the new values of seven negative five and nine. So now again, we want the value that's in the first row or first column, excuse me. And second row, we want this value to be zero. In this case, we have seven. And so to do this, we will take row two and replace it with row one minus row two. So now rewriting our matrix once again, after performing this operation, our first row remains the same. It, its values are unchanged. So we have seven negative five and nine. And now for row two, again, we are taking each value in row one and then subtracting the value in row two. So our new values for row two are all just going to be zero. So we have 00 and zero for row two. And so now with our simplified matrix, we can write out an equation given the information from our first row, we can recall that four hour matrix. The first column or the X coefficients, the second column is the Y coefficients. The vertical line is an equal sign and the last column is the constant. So from our first row, we get the equation seven X minus five Y is equal to nine. And now here we can notice that we only have one row in our matrix that contains nonzero elements. However, we have two variables in our system of equations. So we will need to write the solution using the arbitrary variable Y. So doing this, we just need to solve our equation here for X in terms of Y. So solving for X, we first add five Y to both sides and we have seven, X is equal to nine plus five Y. Now just solving for X, we divide it both sides by the coefficient of X which is seven. And so we find that X is equal to the quantity of nine plus five Y divided by seven. So here we have found our solution for X in terms of Y and said it since it is in terms of Y, we know that the solution for Y is just going to be Y itself. So we can summarize our solutions here by writing our solutions in a solution set where we first have the expression for X which is nine plus five Y all divided by seven. And our second solution is just Y. So here with our summarized solutions, we are left with answer choice D where again we have the solution set of nine plus five Y all divided by seven and just Y itself. 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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