Solve the following set of simultaneous equations using the inverse matrix method, 5x+y=13;3x+2y=5

To solve the given set of simultaneous equations using the inverse matrix method, we need to follow these steps:

Step 1: Write the equations in matrix form. The given equations can be written as:

[ 5 1 ] [ x ] [ 13 ] [ 3 2 ] * [ y ] = [ 5 ]

Step 2: Find the inverse of the coefficient matrix. In this case, the coefficient matrix is:

[ 5 1 ] [ 3 2 ]

To find the inverse, we can use the formula:

inverse(A) = (1/det(A)) * adj(A)

Where det(A) is the determinant of matrix A and adj(A) is the adjugate of matrix A.

Step 3: Calculate the determinant of the coefficient matrix. In this case, the determinant is:

det(A) = (5 * 2) - (1 * 3) = 10 - 3 = 7

Step 4: Calculate the adjugate of the coefficient matrix. To do this, we need to swap the elements on the main diagonal and change the sign of the elements on the off-diagonal. The adjugate of matrix A is:

adj(A) = [ 2 -1 ] [ -3 5 ]

Step 5: Calculate the inverse of the coefficient matrix. Using the formula mentioned in Step 2, we have:

inverse(A) = (1/7) * [ 2 -1 ] [ -3 5 ]

inverse(A) = [ 2/7 -1/7 ] [ -3/7 5/7 ]

Step 6: Multiply the inverse of the coefficient matrix with the constant matrix. In this case, the constant matrix is:

[ 13 ] [ 5 ]

Multiplying the inverse of the coefficient matrix with the constant matrix, we get:

[ 2/7 -1/7 ] [ 13 ] [ x ] [ -3/7 5/7 ] * [ 5 ] = [ y ]

Simplifying the multiplication, we have:

(2/7) * 13 + (-1/7) * 5 = x (-3/7) * 13 + (5/7) * 5 = y

Solving these equations, we find:

x = 3 y = 2

Therefore, the solution to the given set of simultaneous equations is x = 3 and y = 2.