To find the values of x and y for the given linear equations using the inverse and Gaussian methods, we will solve the system of equations step by step.
Using the inverse method:
1. Write the system of equations in matrix form: AX = B
| 3 -2 | | x | | 14 |
| 1 3 | * | y | = | 1 |
2. Find the inverse of matrix A:
A^-1 = 1/(ad - bc) * | d -b |
| -c a |
In this case, the determinant (ad - bc) = (3*3) - (1*-2) = 11
Therefore, A^-1 = 1/11 * | 3 2 |
| -1 3 |
3. Multiply the inverse of A with matrix B to solve for X:
X = A^-1 * B
X = 1/11 * | 3 2 | * | 14 |
| 1 |
= 1/11 * | 3*14 + 2*1 |
|-1*14 + 3*1 |
= 1/11 * | 42 + 2 |
|-14 + 3 |
= 1/11 * | 44 |
|-11 |
= | 4 |
|-1 |
Therefore, x = 4 and y = -1.
Using the Gaussian method:
1. Write the system of equations in augmented matrix form:
| 3 -2 | 14 |
| 1 3 | 1 |
2. Perform row operations to put the matrix in row-echelon form (upper triangular):
R2 = R2 - (R1/3)
| 3 -2 | 14 |
| 0 3 | -3 |
3. Solve for y in the second equation: y = -3/3 = -1
4. Substitute the value of y back into the first equation to solve for x:
3x - 2(-1) = 14
3x + 2 = 14
3x = 14 - 2
3x = 12
x = 12/3
x = 4
Therefore, x = 4 and y = -1.
Both the inverse and Gaussian methods give the same solution: x = 4 and y = -1.
find the value of the variables for the following linear equations using the inverse and Gaussian
methods;
3x – 2y = 14
X + 3y = 1
1 answer