I think that taking a long time to solve this type of problem may indicate a weakness in number sense and possibly algebraic skills, if it is not immediately apparent to the student the type of problem it is and the quickest way to go about solving it.

Really, what you call "number sense and possibly algebraic skills" boils down to what I would call algorithm selection. You need to know how to accurately apply algebraic manipulations to solve these equations using any approach (accept the calculator-only approach). Let's take a closer look at the algorithms I have seen used for this problem so far:

**Main Algorithm Selection:**

M1) Solve one equation for one variable wrt the other and plug that into the second equation. Solve for the second variable. Plug solution back into one of the original equations and solve for first variable.

M2) Multiply one of the two equations by a factor to make one of the two coefficients the same in both equations. Subtract one equation from the other. Solve for the remaining variable. Plug solution for that variable into one of the original equations and solve for the second variable.

M3) Apply linear algebra to the problem to allow for a solution using matrix math. Invert the 2x2 coefficient matrix and multiply it by the 2x1 constant matrix to get the 2x1 solution matrix.

**Secondary Algorithm Selection:**

S1) Do not simplify any fractions first.

S2) Simplify fractions before solving.

**Computation Approach Selection:**

C1) In your head. (Animal)

C2) Using a calculator. (Mineral)

C3) On paper. (Vegetable)

**Figures-of-merit for making these selections (in order of importance):**

F1) Probability of getting the correct answer with this approach. (If you get the wrong answers on the test, your speed does not matter.)

F2) How long does it take to get the answer?

F3) Does the approach give the student a good "sense" of what is going on mathematically. (Frankly, a bit less important during the test, but very important overall.)

Many questions arise when I think about all these choices which must be made by the student:

Q1) How does the student choose these three main algorithms form the MANY algorithms available?

Q2) How many OTHER algorithms are out there to solve these equations? (Rhetorical question: Infinite) How many of these other approaches are BETTER than all of the above? In what ways and cases are they better?

Q3) Is there a "College Board Trick™" that can be applied to this problem? (For instance, is it possible to solve for x*y directly somehow without solving for the individual variables. I don't think there is in this case, but this is something the student needs to consider before starting.)

Q4) Assuming the student knows about these three main algorithms, how do they know which one to choose for this particular problem?

Q5) Why did EVERYONE solve this problem using paper? (DS18 only did it in his head on a challenge.) Why is paper faster in this case? How does the student know a priori?

Q6) Why did Khan Academy use M1->S1->C3 as their approach in the solution provided? This approach certainly gets the lowest score on both F1 and F2. In other words, it is the LEAST likely approach if you want to get the correct answer and it is certainly the SLOWEST solution approach. As a tool for test prep, I find it deplorable that they aren't teaching best practices. How often is Khan Academy teaching inferior techniques? 10% of the time? 50%? 90%? Is this REALLY how we want to teach our children to do algebra?

Q7) Why did DS18 approach this with M2->S1->C3 while DS16 used M1->S2->C3? They both learned the material using exactly same curriculum (Chalk Dust).

Q8) Did Mr. Mosley (in the Chalk Dust videos) teach both M1 and M2? Did he teach both S1 and S2? (I know he does NOT teach skipping steps on paper, at least not initially.) Does he give ANY instruction about how to decide which approach to use when?

Clearly the answer to many of these questions is "practice, practice, practice", but if your practice tool (prep books and Khan Academy) are teaching worst practices, is that really going to get you where you need to be? What if the student is never exposed to M2? (I suspect many are not.)