Solve each system, using the method indicated. 3x + y = -7 x - y ...

Question

Solve each system, using the method indicated. 3x + y = -7 x - y = -5 (Gaussian elimination)

Accepted Answer

Hello. Today we're gonna be using Gaussian elimination to solve the system of linear equations. So we are currently given a system of two linear equations. We have two X minus Y is equal to two and we have X minus Y is equal to three. In order to set up the Gaussian elimination, we're going to be making a matrix that has all of the leading coefficients in each of the equations. Now we have the variables X and Y. So we're going to be making an augmented two by two matrix. Now what's gonna go in the right hand column is going to be the solutions to the current system that's going to be two and three. Then on the left hand side, we're going to be putting the leading coefficients of X and Y in the respective rows. So for the first equation, X has the leading coefficient of two and Y has the leading coefficient of negative one. So we're gonna be writing down two and negative one for the first row. Then for the second row, we have X minus Y which has leading coefficients of one and negative one. So this is going to be our current augmented matrix. And the goal is to go ahead and get this matrix into the form of the identity matrix 1001. And that is going to give us the solutions to the system of equations. So the first thing we wanna do is we want to get the upper left element to be one. Currently, we have one in the bottom left element. So what we can do is we can swap the two rows. So we're going to say that row one is going to be swamped with row two and that is going to give us the matrix of one, negative one and three with two, negative one and two. Now, what we're gonna do is we're gonna get this element to be zero. In order to do that, we can multiply the first row by negative two, then add it to the second row. So the next step is going to be that row two is equal to negative two times row one added to row two. That is going to give us the first row, which is unchanged. So one negative one augmented with three. And since we're changing the second row, negative two times one is going to give us negative two and then plus two in the 1st 2nd row, that is going to give us zero, negative two times negative one is going to give us positive two and two minus one is going to give us one, then finally negative two times three is going to give us negative six and negative six plus two is going to give us negative four. Now, what we wanna go ahead and do is we wanna go ahead and get a leading one in the bottom right element, which we already have. So we wanna make the element right above the one to be zero. In order to do that, we can just go ahead and add the current row two to row one. So the next step is going to be that row one is equal to row two plus row one. And that is going to give us the matrix of the second row which is unchanged. So 01 augmented with negative four and that's going to be zero plus one, which is 11 plus negative one, which is zero and negative four plus three, which is going to give us negative one. So now we have the identity matrix on the left hand side which means that negative one and negative four are going to be the solutions to our system of equations. Or in other words, X is going to equal to negative one, Y is going to equal to negative four. And you can represent this as negative one, comma negative four. So with that being said, the solution to this problem is going to be C so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Solve each system, using the method indicated. 3x + y = -7 x - y = -5 (Gaussian elimination)

Key Concepts

Gaussian Elimination

Augmented Matrix

Row Operations

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