Final answer:
To solve the system of equations using the inverse of the coefficient matrix, represent the system in matrix form. Calculate the determinant of the coefficient matrix and find its inverse. Finally, multiply the inverse of the coefficient matrix by the right-hand side matrix to find the solution.
Step-by-step explanation:
To solve the system of equations using the inverse of the coefficient matrix, we need to represent the system in matrix form.
The given system is:
3x + 4y = -3
x + 3y = 4
Writing the system in matrix form, we have:
[3 4] [x] = [-3]
[1 3] [y] = [4]
To find the inverse of the coefficient matrix, we need to calculate the determinant of the matrix. The determinant of a 2x2 matrix [a b; c d] is given by the formula ad - bc.
The determinant of the coefficient matrix [3 4; 1 3] is (3)(3) - (4)(1) = 9 - 4 = 5.
Next, we find the inverse of the coefficient matrix by applying the formula:
[a b] [d -b]
[c d] = [-c a]/det(A)
Calculating the inverse of the coefficient matrix [3 4; 1 3], we have:
[3/5 -4/5]
[-1/5 3/5]
Finally, we can find the solution to the system of equations by multiplying the inverse of the coefficient matrix by the matrix on the right-hand side of the equation:
[x] = [3/5 -4/5] [-3] = [2]
[y] = [-1/5 3/5] [4] = [4]
Therefore, the solution to the system of equations is x = 2 and y = 4. So, option a is correct: x = 2, y = 4.