-x+y+2z=3
x+2y+4z=1
2x+2y+z=1
By Cramer's Rule:
| 3 1 2 |
| 1 2 4 |
| 1 2 1 |
x = ------------------
| -1 1 2 |
| 1 2 4 |
| 2 2 1 |
(Note: The denominator is the determinant of the coefficients of the x's, y's and z's. The numerator is the same as the denominator except the "x column" has been replaced by the numbers from the right sides of the equations.)
I hope you have learned at least a little bit about resolving 3x3 determinants because it is a bit too involved to explain fully here.
First we will do the numerator:
Traversing the first row:
| 3 1 2 |
| 1 2 4 | = 3 * | 2 4 | - 1 * | 1 4 | + 2 * | 1 2 |
| 1 2 1 | | 2 1 | | 1 1 | | 1 2 |
| 3 1 2 |
| 1 2 4 | = 3 * (2*1 - 2*4) - 1 * (1*1 - 4*1) + 2 * (1*2 - 2*1)
| 1 2 1 |
| 3 1 2 |
| 1 2 4 | = 3 * (2 - 8) - 1 * (1 - 4) + 2 * (2 - 2)
| 1 2 1 |
| 3 1 2 |
| 1 2 4 | = 3 * (-6) - 1 * (-3) + 2 * (0)
| 1 2 1 |
| 3 1 2 |
| 1 2 4 | = (-18) - (-3) + 0
| 1 2 1 |
| 3 1 2 |
| 1 2 4 | = -15
| 1 2 1 |
Now for the denominator:
Traversing the first row:
| -1 1 2 |
| 1 2 4 | = (-1) * | 2 4 | - 1 * | 1 4 | + 2 * | 1 2 |
| 2 2 1 | | 2 1 | | 2 1 | | 2 2 |
| -1 1 2 |
| 1 2 4 | = (-1) * (2*1 - 2*4) - 1 * (1*1 - 4*2) + 2 * (1*2 - 2*2)
| 2 2 1 |
| -1 1 2 |
| 1 2 4 | = (-1) * (2 - 8) - 1 * (1 - 8) + 2 * (2 - 4)
| 2 2 1 |
| -1 1 2 |
| 1 2 4 | = (-1) * (-6) - 1 * (-7) + 2 * (-2)
| 2 2 1 |
| -1 1 2 |
| 1 2 4 | = 6 - (-7) + (-4)
| 2 2 1 |
| -1 1 2 |
| 1 2 2 | = 9
| 2 2 1 |
So:
| 3 1 2 |
| 1 2 4 | -15
| 1 2 1 |
x = ----------------- = ------ = -5/3
| -1 1 2 |
| 1 2 4 | 9
| 2 2 1 |
Now repeat this for y and z:- The denominators for y and z will be the same as it was for x. Don't recalculate it! Just re-use the 9 we have already found it to be.
- The determinant for the numerator of y will be the same as the denominator except the "y column" (in the middle) is replaced by the numbers from the right sides of the equations (just like x's numerator had its first column replaced by these numbers).
- The determinant for the numerator of z will be the same as the denominator except the "z column" (3rd column) is replaced by the numbers from the right sides of the equations (just like x's numerator had its first column replaced by these numbers).
For the numerator of y you should get: 22. This makes y = 22/9. For the numerator of z you should get: -5. This makes z = -5/9.