Solving a Linear Equation with Variables on Both Sides—On the Number Line! #clotheslinemath

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I kid. It would be a bit of an understatement to say that the original video generated some disagreement. And while certainly I'm happy to take responsibility for how I presented it, the tool itself is useful to beginners who are trying to develop the intuitions around inverses and identities that the expert already possesses.

In today's example, I'm extending the number line technique to an equation with variables on both sides. To do so, we adjust our number line slightly. Instead of being a purely numeric tool, we now represent the variable and its multiples on the number line. We can still line up the expressions on the number line based on their equality, and then we can shift to the right or left as is helpful, and scale the number line up or down to make the variable value match what we learn about it.

To be as clear as I possibly can, this is not the one-and-only way you MUST solve all linear equations! In fact, much like we leave behind skip-counting once we grow in mastery at multiplication, I would not expect any expert at elementary algebra to keep using this technique once they understood how the inverses and identities were working together to eliminate terms and reduce the equation to its solution. But for the beginner who does NOT already understand that abstract process, this can be a useful concrete tool.

Thanks for watching!

#clotheslinemath #numberline #algebra

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[IllIlIIIll]

I recently became a high school math teacher and realized that students have a huge knowledge gap. To some, if not many, solving a single variable linear equation may be easy, but many of my students struggle with solving linear equation algebraically. These different methods and representations can really hep. Thanks for your work and I hope to see more amazing work from you.

[phoenixplayz]

While I personally don't like this method, y'all need to stop hating. These types of basic visuals help many people who are new to algebra, and for some reason just can't get the hang of it. This also helps explain why x = 6 in a very simple way. If you don't like the video, don't leave a comment. It's clearly not meant for those who have an average or advanced understanding of algebra. It's to help those who are struggling.

[tamirerez]

And that was a lesson about how to travel from ENGLAND to FRANCE via BRASIL...

[dackid]

Tldr: this is a good skill to have. Keep it up!

Although the method definitely isn't practical, gaining a visual intuition for a problem is a great tool to have. As you are pointing out, this algebra is just a series of transformations on a number line, and that's really cool to visualize.

I've taken many involved classes like topology and abstract algebra (among others), and drawing a picture to see what is going on has been an absolute godsend for understanding theorems and how to prove them.

[Iam_TSA]

Thanks man. I'm getting better at mathematics.

[ari_archer]

why not just math

5x = 3x + 12
5x - 3x = 12
2x = 12 |: 2
x = 6

[Choxchip]

Oh I just subtracted 5x from 3x the I divided 2 on each side and it equaled 6

[Bech]

I don't think I understand. Is the number line to make it more visual for some? All you're doing is -3x and then divide by 2 on each side right?

[pauselab]

ah yes the "easier method". if you want an even "easier" method, draw both graphs and the intersection is the answer. works for every single expression

[Nate.]

I love how you made that original video seem so good because compared to this video it was

[jcool]

I'm confused. Is this how algebra is being taught in schools these days? No wonder my kid stares at me like I'm from Mars whenever I try to help him with his homework.

[davidmurphy]

Um, maybe I'm a bit slow but I'd have done it in R2. Let's relabel 12 as 12y and have the left and side be 5x + 0y.

Or vector2(5, 0) = vector2(3, 12)

Nope, that's a contradiction. Unless you make a matrix to transform the first vector to the second.

Hmmm. Couldn't do it. Blah. Now I really want to know how to do this. I suppose if there are two vectors then there are an infinite amount of matrices which will transform it. Is that true? Given that X and a number are different then surely it follows that it's a 2d geometric problem?

Ah, I'm in over my head. Blah. If anyone ever reads this and feels like helping me I'd be extremely grateful.

[mokied]

Universally beloved 🤣

Okay, so what about:
5x + 3 = 3x + 12

[TheXeneco]

This looks like the most complicated way to solve the given equation…..ever. What’s the fascination with number lines?

[soloeurope]

why would one complicate their life so much by using number lines for this type of question...

[reidflemingworldstoughestm]

Sure, you could do that. Or... you could it a really really _easy_ way instead. The choice is yours.

[coolskeletondude]

what would we do if we added another variable, ex 3x = 2x + y -3
or 5x = 3xy +14 or something
would we add a new line, or change the number line to match the numbers?

[colinjava]

But you could've solved it by the time you'd written out the number line if you just do it the regular way.
This method isn't gonna help when the equations get harder, I personally think writing it using balance scales may make it easier to see than using a number line.
Part of the problem is knowing how to do 5x - 3x, but you can learn its essentially from 5 - 3.

[mikpiotto]

How about saying the common terms add together, bingo you can rearrange the equation. Nothing simple about this and not useful when completing more complex equation. Wouldn't it be best to learn the basics first so they can apply the basics rules to other different equations in the future.