Use the matrix method to solve the system of equations 2x + 4y = 8 and 6x + 3y = -3.
To solve the system of equations using the matrix method, we need to represent the equations in matrix form and apply matrix operations.
Step 1: Write the equations in matrix form:
Let's represent the coefficients of the variables as a matrix called the coefficient matrix, A. And let's represent the variables as a column matrix called the variable matrix, X. The constant terms will be represented as another column matrix called the constant matrix, B.
The given system of equations can be written as:
Equation 1: 2x + 4y = 8
Equation 2: 6x + 3y = -3
In matrix form:
[A][X] = [B]
Where:
A = [2 4]
[6 3]
X = [x]
[y]
B = [8]
[-3]
Step 2: Find the inverse of the coefficient matrix, A:
The inverse of a matrix is denoted as A^(-1).
To find the inverse, we calculate:
A^-1 = 1/det(A) * adj(A)
Where det(A) represents the determinant of matrix A and adj(A) represents the adjugate of matrix A.
Step 3: Calculate the determinant of matrix A:
The determinant is calculated by multiplying the diagonal elements and subtracting the product of the off-diagonal elements.
det(A) = 2*3 - 4*6 = -18
Step 4: Calculate the adjugate of matrix A:
The adjugate of matrix A is obtained by finding the transpose of the cofactor matrix.
The cofactor matrix is calculated by finding the determinant of the submatrices formed by removing each element of matrix A and changing their signs accordingly.
For matrix A,
cofactor(A) = [3 -4]
[-6 2]
The transpose of the cofactor matrix gives us the adjugate of matrix A.
adj(A) = [3 -6]
[-4 2]
Step 5: Calculate the inverse of matrix A:
Using the formula A^-1 = 1/det(A) * adj(A), we can calculate the inverse.
A^-1 = (1/-18) * [3 -6]
[-4 2]
A^-1 = [-1/3 1/3]
[2/9 -1/9]
Step 6: Multiply the inverse of matrix A with the constant matrix, B:
Multiply the inverse of matrix A with the constant matrix B to solve for X.
X = A^-1 * B
X = [-1/3 1/3] * [8]
[2/9 -1/9] [-3]
X = [-1 + 2] = [-1]
[-4 + 1] [2]
Therefore, the solution to the given system of equations is x = -1 and y = 2.