Emily bought 3 books and 2 apples, and she spent £19, while her brother, John, spent £15 on 1 book and 5 apples. What is the cost of one book and one apple?

First we assign, for example, that x = cost of 1 book, and y = cost of 1 apple. Then we make these into simultaneous equations, such as: 3x+2y=19 (eqn. 1) and x+5y=15 (eqn. 2). Now multiply eqn. 2 to get a common factor for x: (2)3 gives: 3x+15y=45 (eqn. 3). Now, we subtract (3) from (1), and eliminate x, so that we can solve for y: (3x+2y) - (3x+15y) = (19)-(45), which simplifies to: -13y=-26. Diving both sides by -13, we get y=2. We substitute this back into equation (1), which gives us: 3x+2(2)=19, expanding: 3x+4=19. Subtracting 4 from both sides, we get: 3x=15, and diving by three, we get x=5. So, the cost of a book is £5 and the cost of an apple is £2.