Linear equations using matrix method
2x-4y=0
3x-2y=3
To solve the system of linear equations using the matrix method, we represent the coefficients of the variables and the constants in matrix form. Let's call the coefficient matrix "A," the variable matrix "X," and the constant matrix "B."
The given equations are:
2x - 4y = 0 ----(1)
3x - 2y = 3 ----(2)
First, let's represent the coefficients in the "A" matrix:
A = [[2, -4], [3, -2]]
Next, let's represent the variables in the "X" matrix:
X = [[x], [y]]
Finally, let's represent the constants in the "B" matrix:
B = [[0], [3]]
Now, we can write the system of equations in matrix form as AX = B:
A * X = B
To solve for X, we need to find the inverse of matrix A and then multiply it with matrix B.
Step 1: Find the inverse of matrix A, denoted as A^-1.
To find the inverse, we can use the formula:
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 2: Calculate the determinant of matrix A (det(A)).
det(A) = (2 * -2) - (-4 * 3) = -4 - (-12) = 8
Step 3: Find the adjugate of matrix A (adj(A)).
adj(A) = [[-2, 4], [-3, 2]]
Step 4: Calculate A^-1.
A^-1 = (1/8) * [[-2, 4], [-3, 2]]
Step 5: Multiply A^-1 and matrix B to solve for X.
X = A^-1 * B
Now, substitute the values and calculate X:
X = (1/8) * [[-2, 4], [-3, 2]] * [[0], [3]]
To multiply the matrices, we multiply the corresponding elements and add:
X = (1/8) * [[(-2 * 0) + (4 * 3)], [(-3 * 0) + (2 * 3)]]
X = (1/8) * [[12], [6]]
X = [[1.5], [0.75]]
So, the solution to the system of linear equations is x = 1.5 and y = 0.75.