Symmetricality
Five equations and five unknowns. Is there an easy way to find the unknown values?
Problem
Symmetricality printable sheet
Here is a set of five equations:
$$b+c+d+e=4$$
$$a+c+d+e=5$$
$$a+b+d+e=1$$
$$a+b+c+e=2$$
$$a+b+c+d=0$$
What do you notice when you add the five equations?
Can you now find the values of $a, b, c, d$ and $e$?
Here is a different set of equations:
$$xy = 1$$
$$yz = 4$$
$$zx = 9$$
What do you notice when you multiply the three equations given above?
Can you now find the values of $x, y$ and $z$?
Is there more than one possible set of values?
Here is a third set of equations:
$$ab = 1$$
$$bc = 2$$
$$cd = 3$$
$$de = 4$$
$$ea = 6$$
Can you find all the sets of values ${a, b, c, d, e}$ that satisfy these equations?
Extension
You may like to have a go at Overturning Fracsum.
Can you create your own set of symmetrical equations?
Student Solutions
Grace from Stowe School, Jettarin from Stephen Perse Foundation School in the UK, Yuk-Chiu from Harrow School in the UK, Ci Hui Minh Ngoc Ong from Kelvin Grove State College (Brisbane) in Australia, John from Calthorpe Fleet in the UK and Kesav solved the first set of simultaneous equations. Kesav explained how the method works:
We add the first set of equations to get $4a+4b+4c+4d+4e=12.$ Simplifying, we can get $a+b+c+d+e=3$. We can subtract the equation $b+c+e+d+e=4$ from the first equation to get $a=-1.$ We repeat the same thing for all of the other equations to get $b=-2,c=2,d=1,e=3.$
This is Ci Hui Minh Ngoc's work, which shows all of the algebra:
Kesav, Grace, Jettarin, Yuk-Chiu and John solved the set of three simultaneous equations by multiplying them all. This is John's work:
Kesav, Grace, Jettarin, Yuk-Chiu and John solved the set of five simultaneous equations by multiplying them all. This is Jettarin's work (click on the image to see a larger version):
Grace came up with a set of symmetrical equations, and then showed how to solve them:
Teachers' Resources
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.