Use Cramer's rule to solve the following:
2x+3x-z=1
4x+y-5z=11
3x-2y+5z=21
To solve the given system of linear equations using Cramer's rule, we need to find the determinants of the coefficient matrix and each of the three matrices obtained when the constant terms in the equations are replaced, one column at a time, by the column on the right side of the equation.
The coefficient matrix of the given system:
| 2 3 -1 |
| 4 1 -5 |
| 3 -2 5 |
The determinant of the coefficient matrix (D) can be calculated as:
D = 2(1)(5) + 3(5)(3) + (-1)(4)(-2) - (-1)(1)(3) - 2(5)(2) - 5(4)(3)
D = 10 + 45 + 8 - (-3) - 20 - 60
D = 10 + 45 + 8 + 3 - 20 - 60
D = -14
Replacing the first column with the column on the right side of the equation, we get:
| 1 3 -1 |
| 11 1 -5 |
| 21 -2 5 |
The determinant of this matrix (Dx) can be calculated as:
Dx = 1(1)(5) + 3(-5)(21) + (-1)(11)(-2) - (-1)(1)(-2) - (-5)(21)(3) - 5(11)(3)
Dx = 5 - 315 + 22 + 2 - (-315) - 165
Dx = 5 - 315 + 22 + 2 + 315 - 165
Dx = -136
Replacing the second column with the column on the right side of the equation, we get:
| 2 1 -1 |
| 4 11 -5 |
| 3 21 5 |
The determinant of this matrix (Dy) can be calculated as:
Dy = 2(11)(5) + 1(-5)(3) + (-1)(4)(21) - (-1)(11)(3) - (-5)(21)(2) - 5(4)(3)
Dy = 110 - 15 - 84 - (-33) - (-210) - 60
Dy = 110 - 15 - 84 + 33 + 210 - 60
Dy = 194
Replacing the third column with the column on the right side of the equation, we get:
| 2 3 1 |
| 4 1 11 |
| 3 -2 21 |
The determinant of this matrix (Dz) can be calculated as:
Dz = 2(1)(21) + 3(11)(3) + 1(4)(-2) - 1(1)(3) - 3(11)(2) - 21(4)(-2)
Dz = 42 + 99 - 8 - 3 - 66 + 168
Dz = 168
Now, we can find the values of x, y, and z using Cramer's rule:
x = Dx / D = -136 / -14 = 68 / 7
y = Dy / D = 194 / -14 = -97 / 7
z = Dz / D = 168 / -14 = -84 / 7
Therefore, the solution to the given system of equations is:
x = 68/7
y = -97/7
z = -84/7