2x plus 5y equals 11 4x - 3y equals 9 solve the following simultaneous equation using matrix inverse method?

Answer 1

2x + 5y = 11

4x - 3y = 9

Since this system of equations, determine a 2 x 2 matrix, it can have an inverse.

A = [a b], [c d] (imagine these as two rows of the matrix A)

A = [2 5],[4 -3]

Since the determinant of that matrix is different from zero (det A = [ad - bc] = [(2)(-3) - (5)(4)] = (-6 - 20) = -26), then it has an inverse (let call it A-1)

A-1 = (1/det A)([d -b], [-c a]) = [d/det A -b/det A], [-c/det A a/det A]

A-1 = [-3/-26 -5/-26], [-4/-26 2/-26] = [3/26 5/26], [4/26 -2/26]

By theorem, x = A-1b is the unique solution of Ax = b, where

x = [x1 x2 x3] , and b = [11 9] (imagine them as columns)

So we have:

x = A-1b = [3/26 5/26], [4/26 -2/26] x [11 9] = [(3/26 x 11 + 5/26 x 9) (4/26 x 11 + -2/26 x 9)] or

[x1 x2] = [(33/26 + 45/26) (44/26 - 18/26)] (imagine them as columns)

[x1 x2] = [(33 + 45)/26 (44 - 18)/26]

[x1 x2] = [78/26 26/26]

[x1 x2] = [3 1]

Thus, x1= 3 or x = 3, and x2 = 1 or y = 1

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