By use of matrices uniquely solve the following system of equations, justifying each step of the calculation: 3x-7y=6, 5y-2x=-3.

The first thing observe is that we are being asked to use a specific approach in order to solve for both x and y, hence we should not approach this question via elimination of a single variable, though it is entirely valid to determine solutions in this way. We transform the system into the form Av=b, where A will be the 2x2 matrix3 -7-2 5,v and b are the vectors (2x1 matrices) with entries x,y and 6,-3 respectively (easier to draw out matrices on whiteboard). We then proceed to calculate the determinant of the matrix A, which provided to be non-zero, will provide a unique solution via the manipulation v=A^(-1)*b.We calculate a determinant of 1 for the matrix A (15-14) and as such get the unique inverse to A as5 -7-2 3,with which we left multiply the vector b to yield the solutions as x = 9 and y =3. To justify the uniqueness of our solution it would be suitable to either state that the inverse of a matrix, where it exists, is unique or to appeal to a graphical explanation, from which some insight into the values of x and y may be gained.