Use Gaussian elimination to find the complete solution to...

Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exist. w + x - y + z = -21 -8w - x + y - z = 14 -w + 6x + y + 6z = -14 Select the correct choice below and fill in any answer boxes within your choice. A. There is one solution. The solution set is { ( , , , ) }. (Simplify your answers.) B. There are infinitely many solutions. The solution set is { ( , , , z) }, where z is any real number. (Type expressions using z as the variable. Use integers or fractions for any numbers in the expression.) C. There is no solution. The solution set is ?.

Transcript

00:01 In this problem, we're given a system of equations more asked to solve it using gaussian elimination.

00:07 So in gaussian elimination, the first step is to go ahead and change it into an augmented matrix.

00:12 So again, you use the coefficients from each of the variables.

00:17 And then separated by line or dash line sometimes you put the constant terms.

00:21 So i've already done that.

00:22 Now, our goal is to get it into row echelon form.

00:25 So that means i need leading ones and then zeros underneath each of those leading ones.

00:30 So i could see that i can get a zero right away in the bottom row if i add the first row and the bottom row together.

00:39 Now i also notice that i've got a lot in between the top and the middle rows that would cancel out if i add those.

00:46 I'm going to go ahead and do that just so that later on there's a little bit less math to do when we switch back to equations.

00:53 Let's go ahead and do this and see where we end up.

00:56 All right.

00:57 So if i add the top and the middle, i get negative 7.

01:02 0, 0, 0.

01:06 And then when i do negative 21 plus 14, i would get negative 7.

01:12 Okay.

01:13 And then in the middle, i'm not going to mess with this at all yet.

01:17 So i'm just going to keep the negative 8, negative 1, 1, negative 1 in 14 that it originally was.

01:25 And then again, i'm adding the top and the bottom rows to get a 0 out front here.

01:30 So 1 plus negative 1, 1 plus 6.

01:33 Negative 1 in 1, 1 in 6, negative 21, and negative 14.

01:39 So that'd give me negative 35 there.

01:43 All right.

01:44 So now i need to still get rid of that negative 8.

01:48 So i notice that my top row, if i divide that by negative 7, so you could think of that as 1 -7th or negative 1 -7th, sorry.

01:59 And then i could multiply it by 8, and i would be able to match that first column to the 8.

02:05 Other row.

02:06 So let's go ahead and do that for the top.

02:09 The bottom, i notice i can divide those by seven.

02:12 I might as well do that.

02:14 There's not a whole big reason to keep it not that way.

02:18 So i'm going to go ahead and do one seventh times the bottom.

02:23 All right.

02:24 Let's see what we get when we do that.

02:25 So again, i'm dividing the top by negative seven.

02:27 That'd give me a one and then multiplying by eight.

02:29 So give me eight.

02:31 Zero, zero, zero, zero stay the same.

02:34 And then i have another negative seven.

02:36 So it would turn into eight just like the first one did.

02:41 Again, i'm going to keep the second row as is.

02:44 Okay, and then on the bottom, multiplying by a seventh, so i'd get zero, one, zero, one, and then a negative five...

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