I think you’re solving these using some creativity. They can actually be solved by rote, using a method where you just have to calculate but don’t have to creatively think about what steps to do. Creativity sometimes enables shortcuts or helps with other, harder problems. I think using a rote method would usually be faster for you, and it’s important to learn the method. It’s also relevant to the general concept of understanding how to follow steps, instructions or flowcharts.
Substitution is mostly better than equation subtraction. Can you do a few problems with substitution, time them, and show your work?
Feel free to look up more educational materials first. I’m not trying to test your current knowledge. But when you actually do the problems, you should be able to do them without looking anything up, so if you do look something up mid-problem, make a note of that.
Okay I’ve done some problems that don’t include fractions using the substitution approach, and using a method that I saw in a few youtube videos. Part of that method is labelling the equations so you know what each bit means.
Yeah, sometimes the time is blown out by stopping to think about if I am getting things right up until that point (because I’m surprised by the results), or catching a mistake while checking my answers in the equations and having to redo bits. These ones that don’t have fractions are easier. But I’ve been doing some practise with fractions and it’s still quicker. Knowing that I should clear the fractions first helps. I was kind of trying to combine steps before but separating them is working better.
can you figure out how to extend it to work with 3 equations and 3 variables?
I think I would:
clear any fractions
rearrange e.g equation 3 for variable a
substitute that into equation 1 and 2 and simplify so they are expressed only in terms of b and c
rearrange e.g equation 2 for variable b
substitute that into equation 1 and solve for c
substitute c into equation 2 and solve for b
substitute b and c into e.g equation 1 and solve for a
It kinda seems like that’d work? It’s kind of hard to tell just typing and I’ll probably figure out more once I try one with more equations. Maybe also there is a better to way to summarise the method. I think you need an equation per variable.
Okay, my first attempt was wrong. I spent 1 hr 45 mins on the problem. I’m not surprised that I got it wrong, but my answers did seem promising as it was developing. I don’t even know where to begin troubleshooting my working. I think I’d just have to try it again along the same path and compare my answers at each step and see if that reveals anything.
My procedure was:
Number all the equations
Rearrange each equation for a different variable
I then realised that that isn’t the right method, and that what I had to do was to rearrange for a variable, and then substitute that into the remaining 3 equations to eliminate it. (It was obvious once doing this that this wouldn’t help me because I’d be reintroducing variables into equations that I should be trying to eliminate them from)
So I then went:
Use equation 3 to get x_1
Substitute x_1 into equation 2 and simplify. that gives me equation 2a
Substitute x_1 into equation 1 and simplify. that gives me equation 1a
Substitute x_1 into equation 4 and simplify. that gives me equation 4a
~now x_1 is eliminated from the remaining system~
Use equation 2a to get x_2
Substitute x_2 into equation 1a and simplify. that gives me equation 1b
Substitute x_2 into equation 4a and simplify. that gives me equation 4b
~now x_2 is eliminated from the remaining system~
Use equation 1b to get x_4
Substitute x_4 into equation 4b and simplify. that gives me x_3
Substitute x_3 into equation 4b and simplify. that gives me x_4
Substitute x_4 and x_3 into 2a and solve for x_2. that gives me x_2
Subsitute x_4 and x_3 into 2 and solve for x_1. that gives me x_1
I think this is important. Judging which one would be easiest to rearrange wasn’t one of the steps in the method. Also, it’s a use of creativity.
I then realised that that isn’t the right method
At this point, you’ve tried out a method and learned something. You’ve reached a conclusion. That’s good. But I think you don’t view it that way. Your goal was more about solving the problem than about using a method (despite the assignment being “follow the method you outlined”).
Changing methods in the middle, improvising, is a common source of error, it’s a use of creativity, and it shows a focus on outcomes not processes.
I think I viewed this as primarily a test to see whether I could do something harder than what I was doing in order to gather info on what problems to next work on. I think you’re right that I wasn’t viewing this primarily as an investigation of the method, despite the assignment being to follow the method I outlined.
The ‘e.g’ part to me meant that I was to judge what was the best one to start with case by case. I used ‘e.g equation 3’ because I was imagining a hypothetical problem where I start with equation 3 for whatever reason. It wasn’t part of the method I outlined to start with the first equation and do them in a particular order. Each time I put ‘e.g’ in that method it meant to me that I was to make a creative judgement about which one to do next.
I can understand why including creative judgements in a method can be bad. Like it might make it harder to error correct the method and harder to use it as a subconscious building block. I think a method that went through each equation in order would be better.
I know that I wouldn’t approach the problem like that again. I did learn that. But you’re saying that my goal was to get the right answer and not to investigate the method, so I might not view what I learned as the success that it is? That makes sense.
In general, when learning things, should I primarily be focusing on methods used, and learning about them, rather than outcomes?
Yes but creativity is useful and hard to avoid for many methods. It’s more often avoidable in math or logic than philosophy or text analysis. In this case the context was about doing it without creativity:
It depends. But I think learning how to use methods is important currently.
Method steps have to be written more explicitly than that to be followable (by others or your future self who doesn’t remember the unwritten information). I didn’t know it meant that.
Assignment: make a flowchart with a non-creative method for solving 1 variable linear equations. If you think you succeeded, then try it out by doing a couple problems following the flowchart steps exactly as written. (You may want to make flowcharts electronically for easier editing in future steps.) 10-40min
I spent all 40 mins on this. I found it hard, and I’m not happy with it because I don’t think it would handle more complex 1 variable linear equations. I also didn’t know how much detail to include. There are parts of the chart missing too that would handle multiple instances of the variable in the equation. I think this could solve a very simple one like 9c + 1 = 10. I didn’t get a chance to try it out on a couple of problems within the time frame.
See if you can find any practical problems with it. One way to look for issues is write down an initial situation (with characteristics like having milk but not having a clean cup), then roleplay it: follow the chart as if you were in that situation and see what steps you would do, in what order, and what final outcome you would get. 5-40min
