Explanation:
Cramer's Method Solves Simultaneous Equations
To solve the simultaneous equations using Cramer's method, we need to find the determinants of the coefficient matrix and the individual matrices obtained by replacing one column with the constant terms. Let's solve each equation separately:
(a) 4x + 3y - 4 = 0; 6x = 8 - 5y
To find the solution, we'll calculate the determinants:
The coefficient matrix is:
| 4 3 |
| 6 -5 |
The constant matrix, obtained by replacing the first column, is:
| 4 3 |
| 8 -5 |
The constant matrix, obtained by replacing the second column, is:
| 4 4 |
| 6 8 |
Now, we'll calculate the determinants:
Determinant of the coefficient matrix (D):
D = | 4 3 |
| 6 -5 |
D = (4 * -5) - (3 * 6)
D = -20 - 18
D = -38
Determinant obtained by replacing the first column (Dx):
Dx = | 4 3 |
| 8 -5 |
Dx = (4 * -5) - (3 * 8)
Dx = -20 - 24
Dx = -44
Determinant obtained by replacing the second column (Dy):
Dy = | 4 4 |
| 6 8 |
Dy = (4 * 8) - (4 * 6)
Dy = 32 - 24
Dy = 8
Now, we can find the values of x and y using the formulas:
x = Dx / D
x = -44 / -38
x = 1.158
y = Dy / D
y = 8 / -38
y = -0.211
Therefore, the solution to the simultaneous equations (a) is x ≈ 1.158 and y ≈ -0.211.
(b) 3x + y = 1; 2x = 11y + 3
We'll follow the same process as before:
The coefficient matrix is:
| 3 1 |
| 2 -11 |
The constant matrix, obtained by replacing the first column, is:
| 1 1 |
| 3 -11 |
The constant matrix, obtained by replacing the second column, is:
| 3 1 |
| 2 3 |
Now, let's calculate the determinants:
D = | 3 1 |
| 2 -11 |
D = (3 * -11) - (1 * 2)
D = -33 - 2
D = -35
Dx = | 1 1 |
| 3 -11 |
Dx = (1 * -11) - (1 * 3)
Dx = -11 - 3
Dx = -14
Dy = | 3 1 |
| 2 3 |
Dy = (3 * 3) - (1 * 2)
Dy = 9 - 2
Dy = 7
Using the formulas:
x = Dx / D
x = -14 / -35
x = 0.4
y = Dy / D
y = 7 / -35
y = -0.2
Therefore, the solution to the simultaneous equations (b) is x = 0.4 and y = -0.2.