# Use Cramer's rule,if possible, to solve the system of linear equations. Algebra!!!?

Use Cramer's rule,if possible, to solve the system of linear equations.

x- y+2z=4

5x+ z=3

x+3y+ z=9

(3, 2, 0)

(2, 0, 3)

(3, 0, 2)

(0, 2, 3)

Relevance

The answer is (0, 2, 3).

Here’s how you get it, using Cramer’s rule: you take 4 determinants.

First, you evaluate the determinant of the co-efficient matrix:

| +1 –1 +2 |

| +5 +0 +1 |

| +1 +3 +1 |.

This we call D, and D = 31.

Next, you replace the first column with the constants, 4, 3, and 9, and evaluate the determinant of that matrix:

| +4 –1 +2 |

| +3 +0 +1 |

| +9 +3 +1 |.

This we call D_x, and we find D_x = 0.

Next, you replace the second column with the constants, 4, 3, and 9, and evaluate the determinant of that matrix:

| +1 +4 +2 |

| +5 +3 +1 |

| +1 +9 +1 |.

This we call D_y, and we find D_y = 62.

Next, you replace the third column with the constants, 4, 3, and 9, and evaluate the determinant of that matrix:

| +1 –1 +4 |

| +5 +0 +3 |

| +1 +3 +9 |.

This we call D_z, and we find D_z = 93.

Finally, we divide D into D_x, D_y, and D_z, to get x, y, and z, respectively:

x = D_x / D = 0 / 31 = 0.

y = D_y / D = 62 / 31 = 2.

z = D_z / D = 93 / 31 = 3.

* * *

In case you are not familiar with the evaluation of 3 X 3 determinants, I will show you an easy method:

| a b c |

| d e f |

| g h i |

is evaluated as follows: D = aei + bfg + cdh – ceg – afh – bdi.

In the case of the first determinant,

| +1 –1 +2 |

| +5 +0 +1 |

| +1 +3 +1 |,

you can write it like this:

+1 –1 +2 +1 –1

+5 +0 +1 +5 +0

+1 +3 +1 +1 +3,

repeating the first two columns, after the third column.

Then, you multiply the three forward diagonals:

+1 X +0 X +1 = 0; –1 X +1 X +1 = –1; +2 X +5 X +3 = 30.

Add these three products and get 0 –1 + 30 = 29.

Next, take the products of the three backward diagonals:

+2 X +0 X +1 = 0; +1 X +1 X +3 = 3; –1 X +5 X +1 = –5.

Add these three products and get 0 + 3 – 5 = –2.

Subtract the second sum from the first sum and get 29 – (-2) = 31.

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