NRICH: The Simple Life

The answer is 5x+8y… What was the question?

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(x5y)
4(2xy)3(x4y)
2(5xy)+3(5x+2y)

Here is a set of five expressions:

(x+y)(x+2y)(x2y)(x+4y)(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 12 5x+912y
3 1 5x+9y
2 32 5x+812y
1 2 5x+8y

Alison’s algebraic approach:

Alison multiplied out the brackets:

a(x+2y)+b(2x+3y){ax+2bx2ay+3by=5x=8y{a+2b2a+3b=5=8=5x+8ya=1andb=2

Age 11 to 14


Algebra & Pre-Algebra, Linear Equations
Middle School, High School, Educator

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