- 3 <em>x</em>₂ + 7 <em>x</em>₃ = 3
<em>x</em>₁ + 2 <em>x</em>₂ - <em>x</em>₃ = 0
5 <em>x</em>₁ - 2 <em>x</em>₂ = 3
(a) I suppose this is asking for the determinant of the coefficient matrix.
Using a cofactor expansion along the first row, this reduces to
(b) Using Cramer's rule, we have
That is, solving for <em>n</em>-th variable consists of dividing [the determinant of the coefficient matrix with its <em>n</em>-th column replaced with the right side of the system, the numbers in boldface,] by [the determinant of the coefficient matrix].
Compute each determinant:
(expanding along the first column)
(again, along the first column)
(first column)
So, we get the solution
(c) Using elimination:
- 3 <em>x</em>₂ + 7 <em>x</em>₃ = 3
<em>x</em>₁ + 2 <em>x</em>₂ - <em>x</em>₃ = 0
5 <em>x</em>₁ - 2 <em>x</em>₂ = 3
Swap the first two equations:
<em>x</em>₁ + 2 <em>x</em>₂ - <em>x</em>₃ = 0
- 3 <em>x</em>₂ + 7 <em>x</em>₃ = 3
5 <em>x</em>₁ - 2 <em>x</em>₂ = 3
Add -5(equation 1) to equation 3:
<em>x</em>₁ + 2 <em>x</em>₂ - <em>x</em>₃ = 0
- 3 <em>x</em>₂ + 7 <em>x</em>₃ = 3
- 12 <em>x</em>₂ + 5 <em>x</em>₃ = 3
Add -4(equation 2) to equation 3:
<em>x</em>₁ + 2 <em>x</em>₂ - <em>x</em>₃ = 0
- 3 <em>x</em>₂ + 7 <em>x</em>₃ = 3
- 23 <em>x</em>₃ = -9
Multiply through equation 3 by -1/23:
<em>x</em>₁ + 2 <em>x</em>₂ - <em>x</em>₃ = 0
- 3 <em>x</em>₂ + 7 <em>x</em>₃ = 3
<em>x</em>₃ = 9/23
Add -7(equation 3) to equation 2:
<em>x</em>₁ + 2 <em>x</em>₂ - <em>x</em>₃ = 0
- 3 <em>x</em>₂ = 6/23
<em>x</em>₃ = 9/23
Multiply through equation 2 by -1/3:
<em>x</em>₁ + 2 <em>x</em>₂ - <em>x</em>₃ = 0
<em>x</em>₂ = -2/23
<em>x</em>₃ = 9/23
Add -2(equation 2) and equation 3 to equation 1:
<em>x</em>₁ = 13/23
<em>x</em>₂ = -2/23
<em>x</em>₃ = 9/23