My esteemed colleague Mark Horley (@mhorley) wrote an excellent blog recently about the balance between the need for understanding when teaching simultaneous equations balanced against ensuring procedures are straightforward enough to support pupils ability to follow (read it here). Reading his reflections led me to reflect on my own approach to simultaneous equations, as well as others I have previously seen, and one that occurred to me literally as I was thinking about them. This blog is designed to act as a summary and chart my journey through the teaching of this topic.

**Elimination**: This is probably the first method I used, and is definitely the sort of approach I was taught at school. Very much a process driven method, I can't remember understanding much about the algebra beyond the idea that I was trying to get rid of one variable so that I could find the other. I find that the subtraction often causes problems (which is partly why Mark's idea of multiplying by -2 instead of 2 is very interesting) and of course the method doesn't generalise well to non-linear equations. I can see this being a popular approach for those people teaching simultaneous equations in Foundation tier.

**Substitution**: Another one from school,

this was the alternative I was taught to

elimination, which was mainly because it

was necessary to solve non-linear

simultaneous equations. I can't remember

it being the method of choice for myself

or any of my classmates, and that is

certainly borne out with my experience of

using it with any other than the highest

attaining pupils.

**Comparison**: Similar to elimination, but for me less

process driven and more focused on understanding the

relationship between the two different equations. This

removes the difficulty around dealing with subtracting

negatives, and allows for the exploration of which

comparisons are useful and which aren't, so it is a little

less 'all or nothing' than the process drive elimination

approach. It also copes nicely with having variables with

coefficients that are the additive inverse of each other, for

example in the pair of equations above if instead of the

approach outlined we multiply the second equation by 3

and get:

4x - 3y = 9 and 6x + 3y = 21

then the comparison would be "the left hand sides have a

total of 10x, and the right hand sides have a total of 30, so

10x = 30."

This is the approach I used when recapping simultaneous

equations with my pupils in Year 11 and they certainly

took to it a lot better than the elimination or substitution

that had used with them the previous year.

**Transformation**: This approach is the

one I have very recently considered, but

not yet tried. The general idea is that you

isolate one of the variables, and then look

at how you can transform that variable in

one of the equations into the other. The

same transformation applied to the other

side of the equation then gives a solvable

equation. Although the equation may be

slightly harder to solve at first, I do believe

this approach has merit. I would suggest

that this approach develops pupils'

appreciation of the algebra and the

relationships between the different

equations in a similar way to the

comparison approach above. I can also see

this approach working for non-linear

equations, like the one below:

etc...

I will almost certainly give this approach a try when I next teach simultaneous equations - when I do I will try and blog the results!

The transformation approach is interesting. I think this topic is one where exposing students to many different "methods" should hopefully lead to deeper understanding. There is a risk that they get overwhelmed, thinking that they need to "learn all the methods". But I'd hope that as they work through them (generally I don't think I even call them anything) with well-chosen examples, they see the logic in it all and generally build confidence in algebra. Of course, neither of us talked about representing the equations graphically. Is that something that comes later, or should we aim to build that from the start?

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