To solve the system of equations using Cramer's Rule, we need to find the determinant of the coefficient matrix and the determinants of the matrices formed by replacing each column with the constants on the right side of the equations.
The coefficient matrix is:
| 1 2 3 |
| 3 -4 -2 |
| 5 3 5 |
The determinant of this matrix is:
det(A) = 1*(-4*5 - (-2)*3) - 2*(3*5 - (-2)*5) + 3*(3*3 - 5*(-4))
= 1*(-20 + 6) - 2*(15 + 10) + 3*(9 + 20)
= -14 - 50 + 87
= 23
Now, we find the determinants of the matrices formed by replacing each column with the constants on the right side:
| 4 2 3 |
| 2 -4 -2 |
|-1 3 5 |
| 1 4 3 |
| 3 2 -2 |
| 5 -1 5 |
| 1 2 4 |
| 3 -4 2 |
| 5 3 -1 |
The determinant of the first matrix is:
det(A1) = 4*(-4*5 - (-2)*3) - 2*(2*5 - (-2)*(-1)) + 3*(2*3 - 4*(-1))
= 4*(-20 + 6) - 2*(10 + 2) + 3*(6 + 4)
= -56 - 24 + 30
= -50
The determinant of the second matrix is:
det(A2) = 1*(2*5 - (-2)*(-1)) - 4*(5*3 - (-2)*(-1)) + 3*(3*(-1) - 5*2)
= 1*(10 + 2) - 4*(15 + 2) + 3*(-3 - 10)
= 12 - 68 - 39
= -95
The determinant of the third matrix is:
det(A3) = 1*(-4*(-1) - 3*3) - 2*(3*(-1) - 5*5) + 4*(3*5 - (-4)*5)
= 1*(4 - 9) - 2*(-3 - 25) + 4*(15 + 20)
= -5 + 56 + 140
= 191
Finally, we can use Cramer's Rule to find the values of x, y, and t:
x = det(A1) / det(A) = -50 / 23 ≈ -2.174
y = det(A2) / det(A) = -95 / 23 ≈ -4.130
t = det(A3) / det(A) = 191 / 23 ≈ 8.304
Therefore, the solution to the system of equations is approximately:
x ≈ -2.174
y ≈ -4.130
t ≈ 8.304
use cramers rule to solve the following 3x3 system of equation in 3 unknown.
x+2y+3t =4
3x-4y-2t =2
5x+3y+5t=-1
1 answer
1 answer