In the last video, we saw what

a system of equations is. And in this video, I’m going

to show you one algebraic technique for solving systems of

equations, where you don’t have to graph the two lines and

try to figure out exactly where they intersect. This will give you an exact

algebraic answer. And in future videos,

we’ll see more methods of doing this. So let’s say you had

two equations. One is x plus 2y is equal to 9,

and the other equation is 3x plus 5y is equal to 20. Now, if we did what we did in

the last video, we could graph each of these. These are lines. You could put them in either

slope-intercept form or point-slope form. They’re in standard

form right now. And then you could graph each

of these lines, figure out where they intersect, and that

would be a solution to that. But it’s sometimes hard to

find, to just by looking, figure out exactly where

they intersect. So let’s figure out a way to

algebraically do this. And what I’m going to do is

the substitution method. I’m going to use one of the

equations to solve for one of the variables, and then I’m

going to substitute back in for that variable over here. So let me show you what

I’m talking about. So let me solve for x using

this top equation. So the top equation says x

plus 2y is equal to 9. I want to solve for x, so let’s

subtract 2y from both sides of this equation. So I’m left with x is

equal to 9 minus 2y. This is what this first equation

is telling me. I just rearranged

it a little bit. The first equation

is saying that. So in order to satisfy both of

these equations, x has to satisfy this constraint

right here. So I can substitute this

back in for x. We’re saying, this top equation

says, x has to be equal to this. Well, if x has to be equal

to that, let’s substitute this in for x. So this second equation

will become 3 times x. And instead of an x, I’ll write

this thing, 9 minus 2y. 3 times 9 minus 2y, plus

5y is equal to 20. That’s why it’s called the

substitution method. I just substituted for x. And the reason why that’s

useful is now I have one equation with one unknown,

and I can solve for y. So let’s do that 3 times 9 is 27. 3 times negative 2 is negative

6y, plus 5y is equal to 20. Add the negative 6y plus the 5y,

add those two terms. You have 27– let’s see, this will

be– minus y is equal to 20. Let’s subtract 27

from both sides. And you get– let me

write it out here. So let’s subtract 27

from both sides. The left-hand side, the 27’s

cancel each other out. And you’re left with negative y

is equal to 20 minus 27, is negative 7. And then we can multiply both

sides of this equation by negative 1, and we get

y is equal to 7. So we found the y value of the

point of intersection of these two lines. y is equal to 7. Let me write over here, so I

don’t have to keep scrolling down and back up.

y is equal to 7. Well, if we know y, we can

now solve for x. x is equal to 9 minus 2y. So let’s do that. x is equal to 9 minus 2

times y, 2 times 7. Or x is equal to 9 minus 14, or

x is equal to negative 5. So we’ve just, using

substitution, we’ve been able to find a pair of x

and y points that satisfy these equations. The point x is equal to negative

5, y is equal to 7, satisfy both of these. And you can try it out. Negative 5 plus 2 times 7,

that’s negative 5 plus 14, that is indeed 9. You do this equation. 3 times negative 5 is negative

15, plus 5 times y, plus 5 times 7. So negative 15 plus

35 is indeed 20. So this satisfies

both equations. If you were to graph both of

these equations, they would intersect at the point

negative 5 comma 7. Now let’s use our newly

found skill to do an actual word problem. Let’s say that they tell

us that the sum of two numbers is 70. And they differ– or maybe we

could say their difference– they differ by 11. What are the numbers? So let’s do this word problem. So let’s define some

variables. Let’s let x be the larger

number, and let y be the smaller number. I’m just arbitrarily creating

these variables. One of them is larger

than the other. They differ by 11. Now, this first statement, the

sum of the two numbers is 70. That tells us that x plus

y must be equal to 70. That second statement, that

they differ by 11. That means the larger number

minus the smaller number must be 11. That tells us that x minus

y must be equal to 11. So there we have it. We have two equations

and two unknowns. We have a system of

two equations. We can now solve it using

the substitution method. So let’s solve for x on this

equation right here. So if you add y to both

sides of this equation, what do you get? On the left-hand side, you just

get an x, because these cancel out. And then on the right-hand side,

you get x is equal to 11 plus y, or y plus 11. So we get x is equal

to 11 plus y using the second equation. And then we can substitute it

back into this top equation. So instead of writing x plus

y is equal to 70, we can substitute this in for x. We’ve already used the second

equation, the magenta one, now we have to use the

top constraint. So if we substitute this in, we

get y plus 11– remember, this is what x was, we’re

substituting that in for x– plus y is equal to 70. This is x. And that constraint was given to

us by this second equation, or by this second statement. I just substituted this x with

y plus 11, and I was able to do that because that’s the

constraint the second equation gave us. So now let’s just solve for y. We get y plus 11, plus

y is equal to 70. That’s 2y plus 11

is equal to 70. And then if we subtract 11 from

both sides, we get 2y is equal to– what is that? 59? You subtract 10 from

70, you get 60, so it’s going to be 59. So y is equal to 59 over 2. Or another way to write it,

you could write that as 59 over 2 is the same thing as–

let’s see– 25– 29.5. y is equal to 29.5. Now, what is x going

to be equal to? Well, we already figured out

x is equal to y plus 11. So x is going to be equal to

29.5– that’s what y is, we just figured that out– plus 11,

which is equal to– so you add 10, you get 39.5. You add another 1,

you get 40.5. And we’re done. If you wanted to find the

intersection of these two lines, it would intersect at

the point 40.5 comma 29.5. And you could have used this

equation to solve for x and then substituted in this one. You could have used this

equation to solve for y and then substituted in this one. You could use this equation

to solve for y and then substitute into that equation. The important thing is, is

you use both constraints. Now let’s just verify that

this actually works out. What’s the sum of these

two numbers? 40.5 plus 29.5, that

indeed is 70. And the difference between

the two is indeed 11. They’re exactly 11 apart. Anyway, hopefully you

found that useful.

I have a review packet in 8th grade for things from 7th grade, but some are new that we’ve never done before, and some are too complicated to remember over summer, so this is a lifesaver

Khan academy is supebbbb……but can you upload more about substitution method. …..plz…please the new one it's my request

Thanks for this video💖😊

I love u

What if the given is in y form?

What do u do when it is like 4x or 18x

This video was very very helpful, Thanks Khan

Thanks best explanation

i have a grade 10 math test and don't know what to do

"pray for me"

Not just useful but entertaining.

I dont get it, from y= -7 turned into y= 7????? How????

What if equations doesn't have a constant term means it is something like this :

A+2b=0,3a+b=0..

Can substitution occur here because I'm getting 0 value for both variables

I get how to do this, but my math teacher wants our systems to be in y= format instead of x=. She explained it on the board but she just acted like everybody knew how to do it and didn’t go into detail. Help??

thankss!!

thanks

You can learn anything🙍🏾😬

Great explanation! I'm glad I was able to find this video.

I DONT GET IT

Who else thinks he sounds like @Marksman

the "sum of 2 numbers is 70, and they differ by 11" problem can simply be solved by a single equation with a single variable like this:

x+x+11=70

2x+11=70, subtract 11 from both sides,

-11=-11

2x=59

x=29.5

the smaller number is 29.5, the larger one is 29.5 +11= 40.5. Done.

Not saying you video is bad, or anything, on the contrary, khan academy is really good, and you explain these things really good as well. Just saying that you could have used an equation that didn't already have a specific way of solving.

THANK YOU I LOVE YOU

I don't know if the people from khan academy know that they are actual gods that have saved so many of us. From the bottom of my heart, thank you.

I slacked off on a project and I dint get this so thanks I’m not gonna get an F now

great way to help explain math to people who struggle in the classroom and struggle to understand it

Bruh this better than my teacher

This helpful

Thank you Khan Academy, you are the reason math is very easy.

x + y = 6

x – y = 2

??? 🙁

Teacher khan, how do we graph??? XD!!!

I have a question, what do you do after you multiply the substituted numbers and then one variable-say like Y- has no other numbers, i mean you have 1 Y with numbers and then you have 1 Y that doesn't have a number…

What's the answer for this equation, i might need help hehehe

3x + 5y = 13

2x + y = 4

Who is here cause they dont understand their math teacher? Thumbs up if you do!

the traditional school system is soo old and makes things so hard, and the teachers only explain once and expect us to know everything, but every student has different needs

I don’t pay attemtion in class so this is Super helpful thx

apparently a logitech mic and 30fps is how i get through math

My teacher Mr.Richards recommended me here, because I had a F, but he knew I was trying. This helped mate! (There is multiple Richard's in the world so I didn't censor it)

Thank you do much, I didn't understand my math teacher but this helped a lot

i want to cry

I still don’t understand I’m really that dumb huh…

Thank you Khan Academy, you have helped me so much

His work was neat

I got a 91 in my Math Exam with he help of this video, thank you!

i hope this guy goes straight to heaven. he helped me through middle school school and still helping me in high school. thank you khan

thank you for helping me understand!

i have a test on all of the systems this morning. i’m at. 57%. wish me luck.

why did you multiply by -1 at the end of the question

Asking my math teacher 100 times is less helpful than your 10 min videos

am sorry sir but apki language smj nahi aaarhi

how do you know when you can multiply by -1 to make the 7 positive, would -7 be wrong?

Sal Khan is a good man. Thank God for Khan Academy.

At 3:18 where did the 11y go? Like he just skipped it I'm confused

AnD I still don’t understand

Helped me a lot thank you very much

I was sleeping when my maths teacher was teaching this so thank you lmao

2:53 why is it negative 27

Wahh i dont urderatand english very much, but it went the thing more perfect that i have already seen♡

Soooooo khan for president or what? I never have wanted to thank someone more, Thank you!

For real though, without Khan Academy I would have failed Math last year and I wouldn't have been in grade 7. Khan Academy's short videos explains better than 45 minutes of Math class everyday. Thank you Khan Academy! <3

thank you for helping me to solve my pbl…..

Oh thank god I found this, I have a test tomorrow and I was full on panicking but now I understand how it works!!

Ok but this much work is two minutes problems a page and I have to do like 10 problems a night meaning what 3 papers a night?

Sir you forgot to add negative on y=-7

3:29

At 3:14 when you gave negative y and add 1 to each side. He put positive 1s on each side . Wouldn't it be negative 1 since the y is negative

This is really helpfulThank you! I have a 83 % in my Honors math class which is a B- so I really needed this!😁

Thanks man!!

I have these two equations:

3x + 9y = -12

12x – 6y = 18

When I follow the method in this video, I get y = 0.714. However, upon checking the correct coordinates, x = 0.714!

I cannot understand why this is happening and where I'm going wrong! Can anyone help?

so how does -6y + 5y just become -y ????

i am going to smash my head into the nearest wall

If this is Algebra Basic..Then how hard would it be when it comes to Algebra Advance?!!

Best teacher ever

Why can’t teachers just do this

why arent you my math teacher