Use Cramer's rule to solve
X1-5x2-2x3=-2
2x1+3x3=11
4x1+x2-x3=1
To solve using Cramer's rule, we first need to find the determinant of the coefficients matrix (D), and then find the determinants of matrices obtained by replacing the corresponding column with the constants matrix for each variable (D1, D2, D3).
The coefficients matrix for this system of equations is:
| 1 -5 -2 |
| 2 0 3 |
| 4 1 -1 |
Therefore, the determinant of D (D1) is:
D1 = | 1 -5 -2 |
| 2 0 3 |
| 4 1 -1 |
D1 = 1(0*-1 - 1*3) - (-5*2*-1 - 1*4*3) + (-2*2*1 - 4*0*-1)
D1 = 0 - (-10 + 12) + (-4 + 0)
D1 = 0 + 2 - 4
D1 = -2
Next, we need to find the determinant for D2. For this, we replace the second column of D with the constants matrix:
D2 = | -2 -5 -2 |
| 11 0 3 |
| 1 1 -1 |
D2 = -2(0*-1 - 1*3) - (-5*11*-1 - 1*1*3) + (-2*11*1 - 1*0*-1)
D2 = 0 - (-55 + 3) + (-22 + 0)
D2 = 0 + 52 - 22
D2 = 30
Finally, we find the determinant for D3. We replace the third column of D with the constants matrix:
D3 = | 1 -5 -2 |
| 2 0 11 |
| 4 1 1 |
D3 = 1(0*1 - 11*1) - (-5*2*1 - 1*4*11) + (-2*2*1 - 4*0*1)
D3 = 0 - (-10 - 44) + (-4 + 0)
D3 = 0 + 54 - 4
D3 = 50
Now, we can use Cramer's rule to solve for each variable:
x1 = D1 / D = -2 / -2 = 1
x2 = D2 / D = 30 / -2 = -15
x3 = D3 / D = 50 / -2 = -25
Therefore, the solution to the system of equations is:
x1 = 1
x2 = -15
x3 = -25
To solve the given system of equations using Cramer's rule, let's first define the coefficients of the variables and the constants in matrix form.
The coefficient matrix (A) is:
| 1 -5 -2 |
| 2 0 3 |
| 4 1 -1 |
The constants matrix (B) is:
| -2 |
| 11 |
| 1 |
Now, let's calculate the determinant of the coefficient matrix (A).