A-REI.5 – what does it mean/look like?

[nvitale]

A-REI.5  “Prove that, given a system of linear equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.”

This came up at a meeting of math specialists in NYC. At first, I was thinking this is just about the idea of elimination to solve a system… or the idea of adding equal things to other equal things will preserve equality. When we started playing around a little more, it seemed deeper. When we replaced one of the equations “by the sum of that equation and a multiple of the other”, we indeed obtained a pair of equations with the same solution. When graphed, the lines intersect at the same point as the original equations, yet the new equation’s line is not identical… the equation is not equivalent to the one it replaced, but we have the same solution.

How would a student prove this? What would a proof  look like? What is the meaning behind this standard?

[Cathy Kessel]

Re “look like”: I checked Illustrative Mathematics and don’t see a task addressing this standard.

Re “mean”: here’s a relevant piece from the Algebra Progression:

Students work with solving systems of equations starts the same way as work with solving equations in one variable; with an understanding of the reasoning behind the various techniques.(A-REI.5)  An important step is realizing that a solution to a system of equations must be a solution of all the equations in the system simultaneously. Then the process of adding one equation to another is understood as “if the two sides of one equation are equal, and the two sides of another equation are equal, then the sum of the left sides of the two equations is equal to the sum of the right sides.” Since this reasoning applies equally to subtraction, the process of adding one equation to another is reversible, and therefore leads to an equivalent system of equations.

[Cathy Kessel]

p.s. The first word in the quote should be “Student” rather than “Students”.

[nvitale]

Thank you Cathy, for your reply!
I do understand that line of reasoning, and it is basically how I began thinking about this. The concept of equality of sums of equal expressions which you and the Algebra Progressions expressed so well is fairly accessible, and is an extension of the reasoning used in solving one-variable equations. I also think the point made about “realizing that a solution to a system of equations must be a solution of all the equations in the system simultaneously” gets to an important point. There is something about the assumption of equality of the equations for the solution set (not equivalence) that makes this reasoning hold water.
What I’m grappling with here is that the resulting system has the same solution as the original system, but the equations are not equivalent (as you would find if you just scale one or both of the original equations, or otherwise manipulate each equation algebraically, but separately) – and the graph would look different (although still cross at the solution point). Another way of thinking about his is that the solution sets of the new equations in the system have the same intersection, but each individual equation’s solution set does not have to be identical to any of the solution sets of the original equations. I’m seeing that this is a result of our assumption of equality is for a special case (for the solution) – our resulting equations are equal (and therefore sum-able or substitute-able) at the solution values, but otherwise are NOT equivalent – interesting!
Still not sure how a students would “prove” this, what proof might look like. I still feel like I’m missing something about this standard – just reading it has made me re-think what I think I know about systems of equations…
Maybe I will post this in the general forum, I would love to hear your further thoughts and thoughts from others.

[Bill McCallum]

I think the discussion shows that you understand this standard very well! In particular, an equivalent system of equations is not necessarily made up equations that are individually equivalent the equations in the original system. A student proof would look something like the discussion here.

[kelicker]

I am still having the same trouble as nvitale. The proof that has been described by the progressions is of simply adding (or subtracting) the two equations together. This produces one new equation that is true. This standard seems to be talking more about the row operations of matrices where you “replace one equation by the sum of that equation and a multiple of the other.” This results in a system of equations where one equation is the same as one of the original equations and the other equation is a new hybrid of the first two original equations. It is asking why this new system has the same solutions as the original.

I do not think this standard is addressing “The Elimination Method” as we know it where you multiply one or both equations by a constant and add them together. This seems like more of a precursor to performing row operations on a matrix.

I myself do not know how to prove this, and I cannot find any proofs online.

[Bill McCallum]

It’s important to remember that we are talking about a system of simultaneous equations at each step, not just about one equation. Your goal is to show that at each step the new system of equations has the same solutions as the old system. Specifically, in this standard, you start with two equations $A=B$ and $C=D$. You want to show that this system of equation is equivalent to the system $A+eC = B+eD$ and $C=D$ (this is the sum of the first equation and $e$ times the second equation). Well, if $A = B$ and $C= D$ then $A + eC = B + eD$ and $C=D$, so any solution of the first system is a solution of the second. And if $A+eC = B + eD$ and $C = D$, then $eC = eD$, so subtracting $eC$ from the left of the first equation and $eD$ from the right, we get $A = B$. And we still have $C =D$. So any solution of the second system is a solution of the first. So the two systems have the same solution and are equivalent.

[kelicker]

Thank you for your response. This is helpful. May I ask, are we interpreting this standard correctly? It is not talking about proving why we can use the Elimination method to solve for a variable, right? Instead it is talking more about why elementary row operations preserve the solution set? Everyone that I have talked to was interpreting this standard as referring to the Elimination method (including, it seems, the Progressions).

[Bill McCallum]

I don’t think I understand the question. You can solve a set of linear equations in many variables by the elimination method. Or, you can represent the system by a single matrix equation and solve by row operations on the matrix. The latter is merely a notational representation of the former; solving by elementary row operations and solving by the elimination method are the same thing at bottom. The use of matrix notation to represent systems of equations is not required in the standards.

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