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Always the Same

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

The Simple Life

Age 11 to 14 Challenge Level:

When Colin simplified the expressions below, he was surprised to find that they all gave the same solution! Try it for yourself.

$$3(x+6y) + 2(x-5y)$$$$4(2x-y) - 3(x-4y)$$$$-2(5x-y) + 3(5x+2y)$$

Here is a set of five expressions: $$(x + y) \quad (x + 2y) \quad (x - 2y) \quad (x + 4y) \quad (2x + 3y)$$

Choose any pair of expressions and add together multiples of each (like Colin did).

Can you find a way to get an answer of $5x+8y$ in each case?

Warning... you will have to multiply the expressions by fractions in some cases.

If you're struggling to get started... take a look below to see how Charlie and Alison thought about the problem when combining multiples of $(x+2y)$ and $(2x+3y)$.

Charlie's trial and improvement approach:

Charlie chose a value for $a$ and worked out the value of $b$ that gave $5x$.
He then kept adjusting the values of $a$ and $b$ until he also got $8y$:

$b$ $a(x+2y) + b(2x+3y)$

$0$ $5x+10y$

$\frac {1}{2}$ $5x+9\frac {1}{2}y$

1 $5x+9y$

$\frac {3}{2}$ $5x+8\frac {1}{2}y$

2 $5x+8y$

Alison's algebraic approach:

Alison multiplied out the brackets:$$\eqalign{a(x+2y)+b(2x+3y)&=5x+8y \\ \Rightarrow \begin{cases}ax+2bx &= 5x\\ 2ay+3by &= 8y \end{cases} \\ \Rightarrow \begin{cases} a+2b &= 5 \\ 2a+3b &= 8 \end{cases} } \\ \Rightarrow a=1 \quad \text{and} \quad b=2$$

With thanks to Colin Foster who introduced us to this problem.