|
Question 168030: I need to solve the equation by using the Gaussian elimination method. I tried it how the book stated but it doesn't look right.
x + 2y - 2z = -2
5x + 9y -4z = -3
3x + 4y -5z = -3
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! x + 2y - 2z = -2
5x + 9y -4z = -3
3x + 4y -5z = -3
-----
you want to get the equations in a form such that the 0 terms and the non zero terms form 2 triangles facing each other as part of a rectangle. the leading non-zero terms form a diagonal.
-----
that would be something like this.
ax.....by.....cz.....d
0......by.....cz.....d
0......0......cz.....d
-----
you can perform 3 basic operations on each equation.
1. you can interchange any rows with each other.
2. you can multiply any row by any constant.
3. you can take any row and multiply it by a constant, and then add to it, or subtract from it, any other row multiplied by a constant.
-----
you do this methodically working from top to bottom left to right.
get a nonzero in the first row of the first column.
then manipulate the equations so that all other rows in that column = 0.
get a nonzero in the second row of the second column.
then manipulate the equations so that all other rows below that column = 0.
repeat the procedure for all remaining rows and columns.
once you're done, you substitute your values into each equation to make sure that all equations have been satisfied.
-----
back go your equation:
x + 2y - 2z = -2
5x + 9y -4z = -3
3x + 4y -5z = -3
-----
since you have a nonzero in the first column first row then we'll leave that row intact for now.
-----
if you take the first row and multiply it by 5 and subtract it from the second row then the second row will become:
first row multiplied by 5: 5x + 10y - 10z = -10
second row before modification: 5x + 9y -4z = -3
second row after subtracting modified first row from it: 0x -y +6z = 7
------
if you take the first row and multiply it by 3 and subtract it from the third row then the third row will become:
first row multiplied by 3: 3x + 6y - 6z = -6
third row before modification: 3x + 4y -5z = -3
third row after subtracting modified first row from it: 0x -2y + z = 3
-----
you now have 3 equations in which the first column is in the form you want it to be in:
those are:
x + 2y - 2z = -2
0x -y +6z = 7
0x -2y + z = 3
-----
next you work on the second column.
in this column you want the third row to be 0.
you could add the first row to the third row but then the first column in the third row would be nonzero so you don't want to do that.
work with the second and third rows only.
multiply the second row by 2 and subtract it from the third fow.
-----
second row multipled by 2: 0x -2y + 12z = 14
third row before modification: 0x -2y + z = 3
third row after subtracting modified second row from it: 0x + 0y -11z = -11
-----
you now have 3 equations with the first column and the second column in the form that you want them.
those are:
x + 2y - 2z = -2
0x -y +6z = 7
0x + 0y -11z = -11
-----
this is the simple gaussian elimination.
you can go one step further and make all the leading coefficients 1.
that i think is what they call the reduced form.
actually your equation is already in that form except for the third row, since the first row x coefficient is 1 and the second row y coefficient is -1.
to make the third row z coefficient 1, you would divided that equation by 11.
it would become 0x + 0y -z = -1
whether you do that or not you will get the same answer.
-----
working with the final modified equations and solving for x,y,z follows:
x + 2y - 2z = -2
0x -y +6z = 7
0x + 0y -11z = -11
-----
work from bottom and solve for z.
-11z = -11
z = 1
-----
go up one equation and solve for y, using the value of z already found.
-y + 6 = 7
-y = 1
y = -1
-----
go up one equation and solve for x, using the value of y and z already found.
x -2 -2 = -2
x -4 = -2
x = 2
-----
looks like the values you want are:
x = 2, y = -1, z = 1
-----
substitute those values in each of the 3 original equations you started with.
3 original equations:
x + 2y - 2z = -2
5x + 9y -4z = -3
3x + 4y -5z = -3
-----
solve first row equation using x = 2, y = -1, z = 1:
x + 2y - 2z = -2
2 -2 -2 = -2
-2 = -2
first equation looks good.
-----
solve second row equation using x = 2, y = -1, z = 1:
5x + 9y -4z = -3
10 -9 -4 = -3
1 - 4 = -3
-3 = -3
second equation looks good.
-----
solve third row equation using x = 2, y = -1, z = 1:
3x + 4y -5z = -3
6 -4 -5 = -3
2 - 5 = -3
-3 = -3
third equation looks good.
-----
|
|
|
| |
