To solve the system of linear equations:
1) Write the system in matrix form: AX = B
where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
The given system:
4x - 6y = -2 ----(1)
x + 3y = 7 ----(2)
Matrix form:
A = [4 -6]
[1 3]
X = [x]
[y]
B = [-2]
[7]
2) Calculate the inverse of matrix A, if it exists.
To find the inverse of A:
1) Calculate the determinant of A, denoted as |A|.
|A| = 4(3) - (-6)(1) = 18 + 6 = 24
2) Swap the positions of the diagonal elements and change the sign of the off-diagonal elements.
A^(-1) = [ 3/24 -(-6)/24 ]
[ -1/24 4/24 ]
Simplifying A^(-1):
A^(-1) = [1/8 1/4 ]
[ -1/24 1/6 ]
3) Multiply the inverse of A by B to solve for X:
X = A^(-1)*B
X = [1/8 1/4 ] * [ -2 ]
[ -1/24 1/6 ] [7 ]
Performing the matrix multiplication:
X = [ (1/8)(-2) + (1/4)(7) ]
[ (-1/24)(-2) + (1/6)(7) ]
Simplifying:
X = [ (-2/8) + (7/4) ]
[ (2/24) + (7/6) ]
X = [ (-1/4) + (7/4) ]
[ (2/24) + (14/24) ]
X = [ 6/4 ]
[ 16/24 ]
Simplifying X:
X = [ 3/2 ]
[ 2/3 ]
Therefore, the solution to the system of linear equations is:
x = 3/2
y = 2/3
Similarly, you can follow the steps to solve the other systems of linear equations using matrix inversion method.
solve the system of linear equations 4x-6y=-2 , x+3y=7 by using matrix inversion method. solve that question and also these
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