When students learn to solve systems of equations, there are two directions my mind goes:
and
Which of these should we be holding up as the most important mathematics for students?
The 8th grade standards introduce systems of equations, and this is a huge challenge for many students. There are a number of ways to solve systems, none of them easy, and all suited for different types of problems. I’m struck in particular by 8.EE.8.b:
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
Solve simple cases by inspection. It seems so trivial, and is easy to skip over on the way to bigger and badder methods of solving systems — but in many contexts where systems are necessary, questions can be solved, or at least approximated, by inspection.
The image above comes from Robert Kaplinsky’s excellent lesson on how much an In-N-Out burger with 100 patties and 100 slices of cheese would cost. It’s really a system of three variables: cheese, patty, and everything else that goes on a burger. Yet there are a number of solution paths that don’t use any of the traditional methods of solving systems.
I think these problems are valuable and meaningful, but I’m not sure how to bridge the gap between solving systems by inspection and solving using traditional techniques. Graphing doesn’t seem like too big a leap, but elimination? It’s a stretch for me.
I have more questions than answers on this topic, and I’ll finish with one more — the menu at the Tuolumne Meadows grill in Yosemite , frequented by long-distance hikers craving calories:
I ate about 2600 calories here on a lunch break while hiking this summer
What questions do you have?
Five Twelve Thirteen 
Hello Dylan
Solve by inspection : The example in the CCSS is not a very sensible one. Here is a better one
x+y=10
x+2y=12
I added another y and the total increased by 2, so y=2
I would always plot graphs until I could see in my head what was going on.
Of course, to plot a graph of 2y-3x=9 requires either knowing about intercepts (both) or doing a bit of algebra to get y=(3/2)x+(9/2)x and using slopes ….
Now you can do some real math with this situation and let them find out that ANY multiple of the first equation added to ANY multiple of the second equation gives another equation going through the point of intersection.
So, pick multiples that cancel (banned word nowadays!) the x terms and get an equation for y only. This then can be seen as “elimination, with reasons”.
Oh, just terminology : A pair of simultaneous equations does not have a point of intersection, only the two graphs have that.
Regarding the real-world usefulness of the burger problem, well it’s fun, isn’t it?
Thanks, Howard. I like your example for solving by inspection (agree the Common Core example isn’t that great; I think they focus too much on whether an equation has one solution/infinite solutions/no solutions). Want to work on scaffolding to build naturally from there to substitution/elimination. Graphing seems to be a bit of a separate idea.
I think the problem being fun is one of its central benefits; I taught a lesson last year from Andrew Stadel around putting post-its on a filing cabinet to deepen understanding of surface area. Is there a real world application of that same principle? Probably, somewhere. Was it more fun to start putting post-its on my filing cabinet, then ask my students how many it would take? Absolutely. And they enjoyed it as well.
What about “How much turf (sod, is it called?) do I need for my lawn?” And what if there is a circular fountain in the middle of it, and “Can we cut the bits up?”. Simple rectangles can get a bit boring. They might suggest replacing the circle by a square (not too good) or by an octagon (much better, need s triangular bits).
I do think that graphing is vital, as visualizing a problem is a big help in moving forward, and I just remembered the simple way!!!
pick a value for x, find the value of y from the equation with the x value substituted
do it again
plot the two points and join them.
This does require that they believe (know)(understand)(appreciate)(pick the politically correct term) that equations like these always graph as a straight line.