yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Linear equations with unknown coefficients | Mathematics I | High School Math | Khan Academy


3m read
·Nov 11, 2024

So we have an equation. It says ( ax + 3x = bx + 5 ).

And what I want to do together is to solve for ( x ). If we solve for ( x ), it's going to be in terms of ( a ), ( b ), and other numbers. So pause the video and see if you can do that.

All right, now let's do this together. What I'm going to do is I'm going to try to group all of the ( x ) terms. Let's group all the ( x ) terms on the left-hand side. I already have ( ax ) and ( 3x ) on the left-hand side. Let's get ( bx ) onto the left-hand side as well. I can do that by subtracting ( bx ) from both sides.

If I subtract ( bx ) from both sides, I'm going to get, on the right-hand side, I'm going to have ( 5 ). On the left-hand side, I have ( ax + 3x - bx ). I could do that in that color for fun: ( - bx ). And that's going to be equal to, well, ( bx - bx ) is just ( 0 ), and I have ( 5 ). It is equal to ( 5 ).

Now what I can do is I can factor an ( x ) out of the left-hand side of this equation, out of all of the terms. So I can rewrite this as ( x \times ( \frac{a}{x} + 3 - \frac{b}{x} ) ), where ( ax \div x = a ), ( 3x \div x = 3 ), and ( -bx \div x = -b ). That’s all going to be equal to ( 5 ).

Now to solve for ( x ), I can just divide both sides by the thing that ( x ) is being multiplied by, ( a + 3 - b ). So I can divide both sides by ( a + 3 - b ). On this side, they cancel out, and I have ( x = \frac{5}{a + 3 - b} ).

And we are done!

Let's do one more of these. So another equation here, we have ( a ) here, we have ( a \times (5 - x) = bx - 8 ). So once again, pause the video and see if you can solve for ( x ).

Well, the way I like to approach these is, let’s just expand everything out. So let me just distribute this ( a ), and then I'm going to collect all the ( x ) terms on one side and all of the non-( x ) terms on the other side and essentially do what I just did in the last example.

So let’s first distribute this ( a ). The left-hand side becomes ( 5a - ax ). That is going to be equal to ( bx - 8 ).

Now we can subtract ( bx ) from both sides. So we're going to subtract ( bx ) from the left-hand side and from the right-hand side. Once again, the whole reason I'm doing that is I want all the ( x ) terms on the left and all the non-( x ) terms on the right.

Actually, since I want all the non-( x ) terms on the right, I can also subtract ( 5a ) from both sides. I'm kind of doing two steps at once here, but hopefully, it makes sense. I'm trying to get rid of ( bx ) here and get rid of ( 5a ) here, so I subtract ( 5a ) there and I'll subtract ( 5a ) there.

Then let’s see what this gives us. The ( 5a )s cancel out, and on the left-hand side, I have ( -ax - bx ). I’m doing that same green color: ( - bx ). On the right-hand side, this is going to be equal to ( -8 - 5a ) (let’s say magenta color).

Now I’ve separated all my ( x )s on one side and all my non-( x )s on the other side. Here I can factor out an ( x ). If I multiply both sides by ( -1 ), I get ( ax + bx = 8 + 5a ).

That just gets rid of all those negative signs. Now I can factor out an ( x ). I get ( x \times (a + b) = 8 + 5a ).

Now we can just divide both sides by ( a + b ). So we divide both sides by ( a + b ), and we're going to be left with ( x = \frac{8 + 5a}{a + b} ).

And we are done! We have now solved for ( x ) in terms of ( a )s and ( b )s and other things, and we are all done.

More Articles

View All
10 HABITS THAT WILL MAKE YOU GREAT | MARCUS AURELIUS | STOICISM INSIGHTS
Everyday each of us fights a battle that the rest of the world knows nothing about. This struggle isn’t with the outside world but within the confines of our own minds. Marcus Aurelius, a Roman emperor and a stoic philosopher, once wrote in his personal n…
Unchaining Captive Elephants in Nepal | National Geographic
I think the most memorable release that I was ever present at is when we put five elephants into a brand new 4-acre Corral. The elephants moved forward by a few feet, all tight together, with the babies underneath them. Then the babies started squealing, …
Free-Tailed Bats: On Location | Hostile Planet
Humans and animals are hardwired to endure, and that includes our “Hostile Planet” crew who had to go through so much to bring you this incredible footage. RENEE GODFREY: We were filming the bat sequence in New Mexico in the middle of the baking hot dese…
Developing strategies for multiplying two digit decimals
Let’s say I want to multiply 3 point 1, or 3 and 1⁄10, times 2.4, which can also be described as 2 and 4⁄10. So pause the video and see if you can do this. Once again, I’ll give you a hint: see if you can express these as fractions. There are a couple of…
Showing My Desk to Adam Savage
Hey, Vsauce. Michael here. The eye is a mirror. When you look into an eye, you can see a small, tiny version of yourself that kind of looks like a doll version of yourself. The Latin word for a little doll is “pupilla.” That’s where we get the word “pupil…
What are affixes? | Reading | Khan Academy
Hello readers! Today we’re going to talk about things called affixes. One of the things that I love about the English language is how flexible its words can be. You can take little word parts and stick them together to make new words. If I read something…