Answer:
To solve the system of equations using Cramer's method, we need to calculate the determinants of the coefficient matrix and the matrices obtained by replacing one column of the coefficient matrix with the column of constants. Let's denote the determinants as follows:
D = |2 1 -3|
|1 -2 2|
|1 1 3|
D1 = |-5 1 -3|
|17 -2 2|
| 4 1 3|
D2 = |2 -5 -3|
|1 17 2|
|1 4 3|
D3 = |2 1 -5|
|1 -2 17|
|1 1 4|
To find the solution, we can use the formulas:
x1 = D1 / D
x2 = D2 / D
x3 = D3 / D
Let's calculate the determinants and solve the system of equations:
D = (2 * (-2 * 3) - 1 * (1 * 3)) - (1 * (-2 * 3) - 1 * (1 * 3)) + (1 * (1 * 1) - 2 * (1 * 1))
= (2 * (-6) - 1 * 3) - (1 * (-6) - 1 * 3) + (1 * 1 - 2 * 1)
= (-12 - 3) - (-6 - 3) + (1 - 2)
= -15 - (-3) - 1
= -15 + 3 - 1
= -13
D1 = (-5 * (-2 * 3) - 1 * (2 * 3)) - (17 * (-2 * 3) - 1 * (2 * 3)) + (4 * (1 * 3) - 1 * (1 * 3))
= (30 - 6) - ((-34) - 6) + (12 - 3)
= 24 - (-28) + 9
= 24 + 28 + 9
= 61
D2 = (2 * (-2 * 3) - (-5 * 3)) - (1 * (-2 * 3) - 17 * 3) + (1 * (1 * 3) - 4 * 3)
= (30 - 15) - (6 - 51) + (3 - 12)
= 15 - (-45) - 9
= 15 + 45 - 9
= 51
D3 = (2 * (1 * 3) - 1 * (1 * 3)) - (1 * (1 * 3) - (-2 * 3)) + (1 * (-2 * 1) - 1 * (1 * 1))
= (6 - 3) - (3 - (-6)) + (-2 - 1)
= 3 - (-3) - 3
= 3 + 3 - 3
= 3
Now, we can calculate the solution:
x1 = D1 / D = 61 / -13 = -61/13
x2 = D2 / D = 51 / -13 = -51/13
x3 = D3 / D = 3 / -13 = -3