Symmetricality

Why do this problem?

The sets of simultaneous equations in this problem have an underlying symmetry which can be exploited in order to solve them more efficiently than by using standard elimination/substitution techniques. Although it is no more difficult than a standard simultaneous equations problem, the unfamiliarity requires students to think creatively. When working on simultaneous equations, it's good for students to see non-standard examples like this one.

Possible approach

This problem featured in the NRICH Secondary webinar in June 2022.

Display the system of equations:

$a+b = 1$
$b+c = 2$
$c+a =-1$
 

"Here are three equations with three unknowns.
Can you find values for a, b and c that solve all three equations simultaneously?"

Give students some time to work with a partner to try possible approaches. They may use trial and improvement, or they may use substitution and elimination.

Once students have had time to tackle the problem, share approaches.

One rather cunning method is to add all three equations together, to give $2a+2b+2c = 2$, so $a+b+c = 1$, and then subtract pairs of equations to find each letter. If no-one comes up with this method, you may wish to show it to them. Then challenge your students to use and adapt this cunning method to solve the three sets of simultaneous equations in the problem.

You may want to mention that there may be more than one solution set for some sets of simultaneous equations.

Key questions

What do you notice when you add the five equations?
What do you notice when you multiply the equations? 

Possible support

Arithmagons and Multiplication Arithmagons invite students to solve a similar system in a context, with fewer variables.
 

Possible extension

Students may find Intersections a thought-provoking challenge on simultaneous equations.