Is there an advantage to teaching solution of simultaneous equations by setting equations equal to each other?

Question

Let's say we're asked to solve the following system of equations for the unknowns $x$ and $y$.

$$x+y=1 \qquad \text{(1)}$$

$$x-y=3 \qquad \text{(2)}$$

My approach, which I'll refer to as serial elimination of variables (SEV for convenience) would be to solve one of the equations for one of the unknowns, substitute that in for that variable in the other equation, solve this equation for the other variable, and then back-substitute to get the other unknown. E.g., $x=3+y$, $(3+y)+y=1$, $y=-1$, $x=2$.

Many of my students seem to have been taught a different procedure, which I'll call SEE for setting equations equal. They would solve both equations for a particular variable, set them equal to each other, solve for that variable, and then back-substitute. So faced with system (1,2), they would do $x=1-y$, $x=3+y$, $1-y=3+y$, $y=-1$, $x=2$.

SEE seems correct but slower. I can think of examples where it doesn't work, e.g.,

$$e^x+xy=1 \qquad \text{(3)}$$

$$\sin y+xy=3 \qquad \text{(4)}$$

You can't even get going with (3,4) by the SEE method, whereas with SEV you can at least reduce it to a single equation in a single variable, and maybe look for a solution by graphing.

I also see the following over and over. Say we have the system (5,6):

$$x+y=0 \qquad \text{(5)}$$

$$x-y=7 \qquad \text{(6)}$$

My students proceed by writing $x-y-7=0$, then setting $x+y=x-y-7$ (setting 0 equal to 0). This hasn't helped, and they don't know what to do next. This seems to suggest that SEE has a disadvantage, which is that it obscures what one is trying to do (eliminate a variable), and this obscurity causes problems for students in applying it.

Why is SEE seemingly so popular? Does it have some advantage that I'm not aware of?

Answer 1

"Why is SEE seemingly so popular?"

The technique I have observed is not just to solve both equations for a particular variable, but to solve both equations for y. There are logical reasons why they might do this.

Most students think of y as a function of x, even if it is not presented that way. Therefore it is natural for them to rewrite both as "y =".

When written as "y=", the students imagine two curves on an x-y plane. At the points where they meet, the curves have the same height and therefore the y for one curve is equal to the y for the other curve. Hence they equate the two y's which they know should tell them the x at which the two curves meet.

Actually, the above sentence is not wholly accurate because many students will not actually equate the two y's and do the algebra. Instead they will enter both equations into their graphics calculator and ask the calculator to find the intersection of the graphs. This is another explanation for the "y=" approach: the only way to get the calculator to graph the curve is to rearrange it into this format. They're used to this and so it's what they do first, even if they are forced to use algebra.

Interestingly, the maths/science/engineering students at my university learn the "adding multiples of equations together" method, and will use it most of the time if presented with raw equations. But if the context where they see the equations is strongly graphical (eg drawing convex sets), they will revert to the "convert to y= and use the graphics calculator" approach.

Even more interestingly, the economics students learn multiple approaches, but their lecturers actually use different ones depending on context, rather than sticking to one. If they have supply/demand curves (whose variables are price P and quandity Q) they will always convert them both to Q= and equate them, despite having shown the students the week before how to use another method for solving simultaneous equations. I think this is because the pull towards the logical story is strong: the equilibrium point is where the quantity supplied is equal to the quantity demanded so equating the two quantities makes sense.

So I think there are two reasons why it is so popular:

• It matches logically to a particular interpretation of the meaning of the equations. That is, at the point where the two curves meet, one of the variables is equal, so you should set one of the variables equal.
• It matches with other approaches they have used already, and they are just making it look more like those other situations so they can use the same approach.

Does it have some advantage that I'm not aware of?

I can see one advantage over your SEV method, which is that there is less scope for errors. You would be aware in yourself how careful you have to be when subbing one expression into another, especially when there are minus signs involved! The SEE method dosn't require this sort of care at the equating stage because you just copy what you have.

Of course, I don't think there is an advantage to only showing them one method. Really you should show multiple methods and a way of choosing which one will work for them today. Almost no method will be foolproof. Personally if I see two linear equations I'll multiply them by things until I can add/subtract to make a variable vanish. But if I see nonlinear equations I'm more likely to make one "x=" or "y=" and sub one into the other. Sure I could conceivably use the same approach for both but I find the first approach easier for linear equations.

Answer 2

From my experience as a user of mathematics, the "SEE" method, making the one-side of two (or more equations) identical and then equating the other sides, is helpful in various instances, for example, when a "non-interesting", or an auxiliary, variable is present.

Assume we are optimizing under equality constraints, say,

$$\max f(x,y) \;\;s.t. \;\;h(x)+g(y) = a$$

The langrangean here is

$$\Lambda = f(x,y) +\lambda [a-h(x)-g(y)]$$

The first-order conditions require

$$\frac {\partial \Lambda}{\partial x} = \frac {\partial f(x,y)}{\partial x} -\lambda h'(x) = 0$$

and

$$\frac {\partial \Lambda}{\partial x} = \frac {\partial f(x,y)}{\partial y} -\lambda g'(x) = 0$$

If we don't really care about $\lambda$ (and in many cases we don't) then the next step is to solve for it,

$$\frac {\partial f(x,y)}{\partial x}\frac 1{h'(x)}= \lambda $$ $$\frac {\partial f(x,y)}{\partial y}\frac 1{g'(x)}= \lambda $$

and equate the left-hand sides

$$\frac {\partial f(x,y)}{\partial x}\frac 1{h'(x)}= \frac {\partial f(x,y)}{\partial y}\frac 1{g'(x)} $$

eliminating from sight the auxiliary variable, the multiplier.

Another case where the SEE method, as a general approach, can be useful to "have in mind", is when one wants to show that two expressions are equal. For example in mathematical statistics, we consider the probability limit of two functions of random variables. In order to check that they are equal, we calculate both separately, and conclude that the left-hand sides are equal if the right-hand sides are equal - we don't equate them directly and try to see whether the equality is true or not.

I understand that the above possibly relate to other levels of mathematics, and the "SEE" method may not help in all cases of solving systems of equations, but I think it is useful for the students to be familiar with this approach also from early on.

Regarding the "SEV" approach, sometimes it produces complicated-looking intermediate steps and I have the feeling it makes plain calculation errors more probable (this does not render it "inferior" of course).

Answer 3

There are multiple ways to solve these "2 equations with 2 unknowns." For your (1) / (2) I'd say they lend themselves to being added up, to 2x=4.

If the variables had coefficients, multiplying through to get the same coefficient on either both X or both Y then let you subtract.

This is also a good time to introduce matrix arithmetic.

Students will use what they are familiar with and comfortable to use repeatedly.

If a method won't work for a certain type of problem you introduce, that's when you offer the reason why and explain the new (to them) method.

Answer 4

I think if I were teaching this I'd probably start with the graphical approach, like, have them put both in slope-intercept y=mx+b form and then find the point of intersection using a graphing calculator, and then from there teach one of those two methods to find the answer. I don't know that your method is necessarily "better" then the other method. Generally I would want students to develop their own methods for things, to have as many different methods as they can in their "toolbox," and to learn how to choose methods that they like to use to solve problems they encounter.

I also find myself wondering if the context would change how students approach this. Like, what happens when you give different problems that require intersecting lines or systems of equations in multiple variables to solve? Does the nature of the problem you give change how students would approach it, absent you giving direct instruction favoring one method over another...?