What is the proper way to represent the solution to a system of two equations

Question

I had a bit of a disagreement with some of my colleagues over the correct way to represent the solution to a system of two linear equations, e.g.:

\begin{align} x + y = 1\\ 3x + 2y = -1 \end{align}

We should find that the solution is $x = -3$ and $y = 4$. I had been telling my students that they should give the solution as:

x = -3
y = 4

(or really anything that put $x = -3$ and $y = 4$ close together to show the solution compactly--as opposed to just finding the $x$ and $y$ in random parts of their work and never summarizing their findings)

One of my students informed me that another teacher told them that they should give the solution as an ordered pair, i.e. $(-3, 4)$. I explained that yes, assuming that we understand that an ordered pair is $(x, y)$, that that could be a way of representing the solution but that I didn't think it was correct and that giving $x = $ and $y = $ is a more correct way of giving the solution.

I further explained that the ordered pair $(x, y)$ represents a point on a coordinate plane. Thus the ordered pair would be a correct answer if the question had stated something to the affect of "If $(x, y)$ represents points on a coordinate plane, then find the point that represents the solution to the given system of equations." I further elaborated that an ordered pair represents a point, so we can get the solution from the ordered pair (e.g. if we were to solve by graphing), but that the ordered pair does not represent the solution--similar to the idea that the x-intercept is a point and we can get the zeros of a function from its x-intercepts but that an x-intercept does not directly represent a zero of the function.

So my question is whether or not it's proper to represent the solution as an ordered pair when asked to solve a system of two equations or are both ways correct since it's pretty well accepted that ordered pairs are of the form $(x, y)$ (or is it even more correct to give the ordered pair rather than $x = $ and $y = $).

Answer 1

A good question to ask. The definitions in play, and the exact wording of the question, are very important. In most algebra contexts that I've seen, the ordered-pair is actually the definition of a solution to a system (see below). My opinion is that this makes sense, because it is the most concise expression of the solution in $x$ and $y$.

Consider carefully different possible English phrasings for a question in this area (which I'm pretty sure need to be included for any well-defined math problem). For example: If the question is "What is the solution set of $x+3=7$?", then a response of "$\{4\}$" precisely answers that question, and is commonly used (similar to answering, "What color is the sky?" with the one-word response, "Blue"). Note that this differs from a direction of: "Solve $x+3=7$", to which the correct response is indeed the assertive statement "$x = 4$". Likewise, if the question asks for the solution to an independent system of equations, then the ordered pair is perfectly acceptable, and preferable by virtue of being more concise.

Certainly a naked $4$ should not be any line of the mathematical work (even if it is a correct response to the first English-language question above). The algebraic writing should always be fully-formed statements involving some relation symbol (most commonly equals), or proper English sentences (or whatever is the local language).

Be aware that the use of ordered pairs for system solutions is in fact the standard definition in most algebra books and standardized tests that I see, for example: Martin-Gay Introductory Algebra, Bittinger Intermediate Algebra, Sullivan College Algebra, Ratti & McWaters Precalculus, etc.

"A solution of a system of two equations in two variables is an ordered pair of numbers that is a solution of both equations in the system." -- Martin-Gay, Sec. 7.1.

"A solution of a system of two equations in two variables is an ordered pair that makes both equations true." -- Bittinger, Sec. 3.1.

"A solution of a system of equations in two variables $x$ and $y$ is an ordered pair of numbers $(a, b)$ such that when $x$ is replaced with $a$ and $y$ is replaced with $b$, the resulting equations are true." -- Ratti & McWaters, Sec. 8.1

In each of these texts, all exercises in systems of equations have their solutions given as ordered pairs (or triplets). Assuming that you're working in conjunction with some textbook, then I would encourage you to read carefully and follow the practice of the book being used in the course.

Answer 2

I've seen it both ways. Students calculate the value of x and write "x=3" and then calculate y. Some teachers will accept this, others will subtract a point out of the 5 the question might be worth.

The better teachers will be explicit in how they require an answer. Similar to the instruction "rationalize the denominator" "leave in radical form" etc.

Answer 3

I think it doesn't matter for linear systems. However, for non-linear systems it's a different matter.

Consider the equations $x^2+y=3$ and $y^2-x^2=-1$. It would be ambiguous to write $x=\pm\sqrt{2},\pm\sqrt{5}, y=1, -2$ because the values $x=\sqrt{2}, y=-2$ do not satisfy the given system. Using ordered pairs $(x,y) = (\pm\sqrt{2},1), (\pm\sqrt{5},-2)$ is clearer.

Perhaps using ordered pairs for linear systems instills "good practice" for the future.

Answer 4

This began as a comment, but became long-winded enough that I decided to post an answer:

A wonderful thing about mathematics (and being a mathematics teacher) is that the proper form for students to give any symbolic solution is that in which they are prompted, and they should be conditioned both to identify a form that is appropriate for a given problem and give any answer in a variety of ways if called upon to do so.

One thing that I try to communicate to my students is that symbols and notations are arbitrary. In fact, this is really a "eureka" moment in mathematics education: when students realize that a variety of systems which bear wildly different mathematical descriptions can be solved using a single algorithm. In addition, many measurable physical models which vary wildly in derivation and application bear identical mathematical representation. Really, for students to understand these and apply them in general is the end of our instruction.

For this problem specifically, to give an answer as a point $(x_o,y_o)$, the intersection of vertical and horizontal lines $x =x_o , y=y_o$, a vector $\langle x_o, y_o\rangle$ or in any other form should depend on how the question was posed and any geometric considerations that have been given.

Consider this a teachable moment- ask them why they gave their answer in a certain form and if they think there is a better way in subsequent questions. This is especially useful in geometry courses that "skirt around" definitions of objects like points. You might be surprised at some of the insightful answers you get- I usually am.

Answer 5

A good way to avoid confusion could be to say "The solution $(x,y)$ is ..." and "The solutions $(x,y)$ are ...".