Question

Solve each of the following sets of simultaneous equations using the matrix method:

(a) 3 x+5 y=7, x+y=1

Answer 1

To solve the simultaneous equations 3x + 5y = 7 and x + y = 1 using the matrix method, form matrices with coefficients, variables, and constants, find the inverse of the coefficient matrix if possible, and multiply it with the constants matrix. The solution is x = -1.5 and y = 2.

Step-by-step explanation:

To solve the set of simultaneous equations 3x + 5y = 7 and x + y = 1 using the matrix method, we first need to express the equations in matrix form. In this case, the coefficients of the variables form the matrix A, the variables form the matrix X, and the constants form the matrix B.

The matrices for these equations are:

A = [[3, 5], [1, 1]]
X = [[x], [y]]
B = [[7], [1]]

To find X, we calculate A-1B. First, we find the inverse of A (if it exists).

1. Calculate the determinant of A: det(A) = 3*1 - 5*1 = -2
2. If det(A) ≠ 0, A is invertible. Since det(A) = -2, we can proceed.
3. Find the inverse of A by following the formula A-1 = (1/det(A)) * adj(A), where adj(A) (the adjugate of A) is the transpose of the cofactor matrix of A.
4. The inverse of A is A-1 = (1/-2)*[[1, -5], [-1, 3]] = [[-0.5, 2.5], [0.5, -1.5]].
5. Multiply A-1 by B to find X: X = A-1B.
6. The result is X = [[-0.5, 2.5], [0.5, -1.5]] * [[7], [1]] => X = [[-0.5*7 + 2.5*1], [0.5*7 - 1.5*1]] => X = [[-1.5], [2]].

Therefore, the solution to the set of equations is x = -1.5 and y = 2.