00:01
For this problem, we're given three different systems of equations, and we're asked to solve them using gaussian elimination.
00:06
So remember the first step for gaussian elimination is to go ahead and set up an augmented matrix using the equations you were given.
00:17
So remember, any coefficient that is missing, but you saw the letter is a one.
00:21
If there is no letter there at all, so like the variable is completely missing, that means that the coefficient would have to be a zero.
00:31
So if i pull over the coefficients in the constant terms, this is the augmented matrix that i get.
00:39
I do a solid line here to separate my constant column from the rest of my matrix.
00:47
A lot of times you'll see a dash line that is also perfectly acceptable.
00:51
This is just easier for me to do and helps me keep organized.
00:54
All right.
00:55
So in gaussian elimination, we are working on getting this matrix change to row echelon form.
01:00
So that means i will want ones in my leading, in my main diagonal here.
01:08
And then everything under it over here will need to be zeros.
01:13
So to do that, i need to add and subtract and multiply the rows until i can get those zeros first and then go after the ones.
01:20
So i can see right away that i can get a zero in my middle row in the first column if i add the top two rows.
01:27
Now, i can't do anything quite yet to get rid of that bottom.
01:31
Two there in the last row, first column.
01:34
So i'm going to go ahead and double my first row, multiply it by two, so that in the next step, i can subtract those two rows.
01:41
So if i double the top, i get two, negative four, six, and eighteen.
01:50
For the middle, when i add them, i get zero, one, three, five, and the bottom is just staying the same for this time.
02:00
All right, so now i can go ahead and get my zero in that bottom row if i add sorry subtract the top two row or sorry row one minus row three and then i can divide the top back to what it was so i can either divide by two or multiply it by half however you want to think of that so my top row is just returning to what it was one negative two three and nine i'm going to leave my middle row as is zero one three, five.
02:32
And then top row minus bottom row.
02:34
Two minus two, negative four minus negative five or negative four plus five, six minus five, and 18 minus 17.
02:42
Right.
02:43
So now i just need a zero in the bottom.
02:45
And i see if i do my second row minus my third row, i will get that.
02:50
So let's go ahead and do that.
02:51
Again, i'm leaving my top two rows alone.
02:57
That's one thing about the matrices is that you rewrite them frequently.
03:01
So be very careful when you're copying over.
03:04
A lot of times that's where students will make a mistake.
03:07
All right.
03:07
And then row two minus row three.
03:09
So one minus one, three minus one, five minus one.
03:13
Then all it's left is to change that bottom two into a one.
03:16
So i'm going to multiply it by a half, or you could think of it as dividing by two, whichever works better for you.
03:22
So one, negative two, three, and nine are the top.
03:28
Zero, one, three, and five.
03:30
And then zero, zero, zero, one and two.
03:34
All right.
03:34
So now we are going to go ahead and change back to equations.
03:39
So x minus 2 y plus 3z equals 9 is what the top row gives me.
03:46
And then y plus 3z equals 5 is what the middle one gives me.
03:52
And z equals 2 is what the bottom one is.
03:55
And go ahead and write the answer as an ordered triple so that you can see it kind of in a nice organized way.
04:00
So z i know is two.
04:03
So i already know one of them.
04:04
I just need to find the other two.
04:06
So i'm going to go ahead and plug z of two into the previous equation, the y plus 3z.
04:12
So i replace the z with the two, and then i'm going to finish solving that.
04:16
So i plus six equals five.
04:18
If i subtract six from both sides, i get y equals a negative one.
04:22
And i can plug both the z and the y into the top equation.
04:27
So x minus two times my negative one, which is what i know y is, plus three.
04:35
Times z which is two equals nine.
04:38
So i'll get x plus two plus six equals nine.
04:43
So x plus eight equals nine.
04:45
Subtract eight from both sides and i get my x equals one.
04:49
So there is my order triple for a.
04:52
We just repeat the process with the other two matrices to solve those.
04:58
All right.
04:58
So here's the second problem.
04:59
I went ahead and set up my augmented matrix.
05:01
Be careful of that missing z in the top row.
05:04
Make sure it's zero.
05:06
All right.
05:06
So then we're going to go through the process again of getting those zeros and then focusing on the ones so i notice that i can't really subtract anything in that first column i can maybe work in the second one but i find it easier just to work left to right so i'm going to go ahead and match everything in my first column to eight so i multiply the top by four multiply the middle by two and the bottom is already there right so when i do that i get eight 12 zero zero and then in the middle i'll get eight six six negative 2, 0, and the bottom just stays as it's 8330.
05:44
All right, so now i can go after those zeros in the first column.
05:47
So i'm going to do r1 minus r2 to get the middle.
05:50
In r1 minus r3, you could do like r2 minus r3 for one of those if you wanted to, but i'm just going to go ahead and do it this way...