Gaussian elimination

Question

solve using gaussian elimination

x+3y+4z=1

3x+4y+5z=3

x+8y+11z=2

Answer 1

 X   Y    Z     C

 1   3   4      1

 3   4   5      3

 1   8   11    2

 

First, do -3*row1 + row2  and -Row1 +  row3

X    Y    Z    C
1    3    4    1
0    -5   -7   0
0     5    7   1

 

Finally, add rows 2 and 3 and store the result in row 2

1  3    4    1

0 -5   -7    0

0   0    0    1

 

The system has no solution as the last equation reads 0 = 1.

It is an inconsistent system.

 

To see this more clearly, do this without the matrix.

First pair the top two equations and eliminate x:

 

x   + 3y + 4z = 1   <--- mutiplies by -3, then adds

3x + 4y + 5z = 3

 

 

-3x - 9y  -12z = -3 
3x + 4y + 5z = 3

 

    -5y - 7z = 0  <--- remember this equation as equation A ***

 

 

Pairs up the last two equations and eliminates x.

 

3x + 4y + 5z = 3

x + 8y  + 11z = 2  <-- multiplies by -3, then adds

 

3x + 4y  + 5z = 3

-3x + -24y + -33z = -6

 

-20y + - 28z = -3  <--- remember this equation as equation B ***

 

 

putting equations A and B together:

-5y - 7z = 0

-20y + -28z = -3

 

Notice that equation B is exactly 4 times equation A. So multiplying equation A by -4 and the adding:

 

20y + 28z = 0

-20y + -28z = -3

 

0y + 0z = -3

0  + 0 = -3

0 = -3 <-- game over. system has no solution