You may also like

problem icon

Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

problem icon

Calendar Capers

Choose any three by three square of dates on a calendar page...

problem icon

Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

The Simple Life

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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 $8y$:


$a$
 
$b$ $a(x+2y) + b(2x+3y)$

$5$
 
$0$ $5x+10y$

$4$
 
$\frac {1}{2}$ $5x+9\frac {1}{2}y$

$3$
 
1 $5x+9y$

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

$1$
 
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.