Solve the simultaneous equations 4x + 5y = 13 and 3x - 2y = 27.

Using the elimination method to remove the variable x, we need both equations to have the same values of x. To do this we find the lowest common multiple of the x values: the lowest common multiple of 3 and 4 is 12. Multiply equation (1) by 3 (so that the x value equals 12) to give 12x + 15y = 39. Multiply equation (2) by 4 to give 12x - 8y = 108. Subtract equation (2) from (1): 23y = -69. Divide both sides by 23 in order to find the value of y: y = -3. Use the value of y that we have found and substitute it back into either equation (1) or (2) in order to find the value of x. Taking equation (1), wherever there is a y replace it by the value (-3): 4x + 5(-3) = 13, and now we can solve for x. 4x - 15 = 13 -> 4x = 28 -> x = 7. Therefore, the solution to this pair of simultaneous equations is x = 7 and y = -3.