Chapter 2: Linear equations in one variable

# Solving equations with the variable on both sides

## How to solve an equation with variables on both sides (20 - 7y = 6y - 6)

Sal solves the equation 20 - 7y = 6y - 6.
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## How to solve an equation with variables on both sides (20 - 7y = 6y - 6)

Discussion and questions for this video
What if there are more than 1 "y" like 3 + 3y + 5y = 4 - 2y + 3y? Would you combine the y's?
you combine the y's to make the equation easier to work with, and so you can reduce messing up, and it takes less steps. does this help?
1 Comment
What is a co-ofichant?
A coefficient is the number in front of a variable that tells how much of the variable is contained in the problem. For example, in the expression 2x, two is the coefficient. There are two x's, which can also be represented by x+x.
1 Comment
How come he divides by 13 in the end?
He divides both sides by -13 to come up with only one y on the left side. -13y = -26 is not the full answer that we need, we are looking to find what the value of only 1 (one) y is and that is why we need to divide both sides by -13.

You can also multiply both sides by -1\13 (multiplicative inverse of -13) and you will get the same answer !

Divide both sides by -13:
-13y = -26
-13y/-13 = -26/-13
y=2

or multiply both sides by -1\13:
-13y = -26
-1\13 * -13y = -26 * -1\13
y=2
Does anyone know why he divides by 13 in the end
So then he can get the y value by its self and your answer ends up being y=2
is there any other way to solve this equation?
It's more confusing until you get used to it, but some write in the operations (say, adding 2x or dividing by 3) inside the existing line's equation. This becomes much more common at college level where no one wants to write 50 lines for simplifying to 10a -3b + 7c + 2x + 4y - 5z = 8 from where there's likely a bunch of variables on a side you don't want them to be. You just do several subtractions/additions at once and then divide the entire equation by the coefficient. It gets really burdensome when you're dealing with a dozen terms or more to do it one term at a time! If you want an actual example of doing multiple terms at once, I can probably find one for you or you can just look up precalc/trigonometry videos to see it in action.
How do you know or decide which side of the equation to put the variables on?
There is no difference whatsoever. Right or left works. Depends on what you are comfortable with. Generally you will move it to the left, unlike Sal did here, but either way's fine. Like Caroline said, you read left to right. Just do what you are comfortable with. Both sides will come out the same.
how do you this problem? y-11=4/5(x-13)
i dont get how you come up with 2 y insted of 1 y
I have a question that needs solving what do i do for
-7 - 3z = 8 + 2z
When I look at an equation like this, I like to keep the variable positive, so when I see -3z on one side and 2z on the other, I want the z to have a positive coefficient in front of the variable. To do that, I'm going to *add* 3z on both sides of the equation because if we subtracted 2z from both sides, the coefficient on the left side of the equation would be -5z. So, we add 3z on the both and you should get -7 = 8 + 5z.

We want to get z on one side of the equation, so to do that we'll have to work with the order of operations. Instead of starting with multiplication like you would expect, we are going to start at the addition/subtraction part of the order of operations because we are trying to get a variable by itself. We start by *subtracting* 8 on both sides of the equation and we should come up with -15 = 5z.

Now the problem looks pretty straight forward. Just divide 5 on both sides to get z by itself. The answer is -3 = z.

Hope this helped.
Do the variables have to go on the left side ?
No, variables can go on either side. Left side is preferred though, because you read left to right. Right or left, it doesn't matter, both will come out with the same answer.
how do i solve 24=6(-x-3)
Well, to get you started, you would use the distributive property and get:

24=-6x-18

Hope this will help!

Imagineer ;)
Can't you not do the subtraction and just move numbers with or without variables to the other side? Would that not get you the answer as well? For example, with
20 - 7y = 6y-6, you could move -7y to the other side so that it becomes positive and equals 6y + 7y which also equals 13y. And then you could move -6 to the other side so it also becomes positive and adds up to 26. Then 26 = 13y is simplifiable into 2 = y. Is that not possible as well?
Abhijeet,
Yes, you can definitely "just move it to the other side and change the sign".

But when you do, what you are really doing is subtracting (or adding) the same thing to both sides, and then using the identity property of addition to make the expression subtracted from itself to become zero and disappear.

It is important to understand the concept of adding or subtracting the same thing from each side which allows the expressions on each side remain equal.

Once you fully comprehend the concept of doing the same thing to each side, you can then skip the intermediate steps and it will look like you just move the expression to the other side while reversing the sign.
How do you solve -5-x=2x-(4x+6)
use a calculator
At 0:22 do you have to add 7y or subtract 6y?
Can't you add 6 to both sides?
Yes, you can do either, add 7y to both sides or subtract 6y from both sides, it doesn't matter. If you subtract 6y instead of adding 7y you will get :
20-7y=6y-6
subtract 6y from both
20-13y=-6
subtract 20 from both sides.
-13y=-26
divide both sides by -13
y=2
You would not add 6 to both sides, 6y is positive, you do the opposite operation when you move a number to the other side of the equation, you would subtract 6y from both sides.
how to solve a variable with decimal?
Same way how u solve that. :)
Who is Alyana F.???
how should i know that
Is'nt there a certain problem where the answer is unknown?
Yes, sometimes there is no answer. But only if the equation has variables on both sides.
why is a negative divided by a negative a positive
At 3:23, couldn't have Sal divided/multiplied by -1 before dividing by 13? That way, it would all be positive without dividing by negatives, which is hard for some (not including me) to understand, therefore making this lesson easier?
You could do that - but because meany do find dividing by negatives troublesome, Sal did it this way on purpose. In the end, dividing 2 negatives is a skill that must me mastered as soon as possible, or it will be the cause of numerous errors later on.
What is a coefficient?
A coefficient is the constant that is multiplied by a variable.
For example, in 2x + 7 the coefficient of x is 2
And in x/5 the coefficient of x is ⅕
And in 6x + 7y the coefficient of x is 6 and the coefficient of y is 7
1 Comment
How do you solve this equation e/5 - 6 = -9
Anthony is correct in his answer, but let me show you how he got it.

You would approach the problem as follows: First, add 6 to both sides. The equation now reads e/5 = -3. Next, multiply both sides by 5, which leaves us with e = -15.

Is it better to start on the right or left. Or does it matter?
It doesn't matter, for some people its easier for them to work out the problem if its on the left side. And some prefer it on the right side.
Thank you my teacher couldnt explain dat
u may wanna spell that right "dat" sounds like a gangster speaking :)
What is a coefficient?
A coefficient is the number that is being multiplied by a variable. For example in the term 5x, 5 is the coefficient.
What does y stand for?
Y is a common variable used to define one of the axes, as well as one of the ordered pairs.
At 0:20 to 0:28, why is it much easier to keep the variables on the right side?
It's personal preference, there's no official "easier" side. :)
Happy holidays!!
Does anyone know a video for equations with 2 variables on both sides?
do you have to always subtract with the first number before doing the other numbers?
generally but it can differ.
you are doing gods work man. keep it up =D
No, adding 6y would not cancel out one of the y's or in other words a y variable would still be on both sides of the equation
It would be 20-y=12y-6 after adding 6y
where do I go to practice these problems on Khan Academy?
yes.....its called transposition
it means that you can take term(6y is a term, so is -6) and move it to the other side of the equation., but you have to do the opposite operation with it.
eg: 20-7y=6y-6
20-7y-6y=-6
20-13y=-6
-13y=-6-20
-13y=-26
y=-26/-13
y=2
hope that helped
How do you know which number to take away? For example, when Sal added 6 to the right side. How do you know to do it on that side? I'm really confused on this, please help!
because, -6+6=0.if we add some other to -6 it would give us some other no.
In what case in real life would we get a question with variables on two sides?
It's good to know encase of exchanges. For example, if the CEO of a company wants 6 pears and $4 for 7 apples and$2, is it a worth exchange? If your apples are more than \$2 each, it would most likely be worth it. Hope this helps! :D
were do i go to practice this?
I cant find any Multi step equations 1 that i can actually do. need help
How bout 10y(4x+5)3y
at 0:41 Why was he adding 7y on both the sides?
To change the equation so that there is only one y-expression.

He started with 20 - 7y = 6y - 6

Adding 7y to both sides makes 20 - 7y + 7y= 6y - 6 + 7y

(We know that the two sides are still equal to each other because has done the same thing to both sides.)

Simplifying, he gets 20 = 17y - 6

He chose to add 7y to both sides so that the left-hand side became 20, an expression not involving y.
Why would you start adding or subtracting on a certain side instead of the other? I get confused a lot for where i put the distributive variable and/or the symbol (- minus)when adding the opposite or even when you decide to multiply instead of divide or vice versa. Is there a rule for that? Say you have 7y+(-6y)+5 =7x+8x-4 i'm not sure if that would even work but it is just a example. *if using a certain law can you describe how that law is used with the problem and why you wouldn't use another law?
@modernmarvelz No problem
x or any letter is left, and non-letter is right, right?
Brian,

Technically it doesn't matter. To solve for an unknown the unknown variable could be on the left or on the right, as long as when you are finished it's ISOLATED (or "all by itself"). x=3 is the same thing as 3=x (that's called 'the symmetric property of equality'), so if you happen to isolate the variable on the right, you are allowed to reverse the direction of the equality. And you don't even need to wait until you've isolated everything to do the switch. For example, if you have 23=4x+2 you could write 4x+2=23… that's that same "symmetric property of equality" at work.

Now that said, out of habit most people tend to move the unknown to the left side and everything else to the right side. I think it just sounds better to say "x=3" than to say "3=x," but really all you need to do is isolate the unknown on ONE side. It doesn't specifically have to be the LEFT (although most people do chose to put the letters on the left!).

But in general, your basic approach is correct. You've got to isolate the variable to solve for it.
What is a coefficient?
*A number multiplied to a variable in an expression* is a coefficient.
For example in 5x + 2y + 3 = 0 , 5 is the coefficent of x and 2 is the coefficent of y.
were is this usefull in life what job does this beside math teacher
This is a question that comes up a lot, and there are many different answers. Here are just a few (not all mine, just general answers which are often given).

- If you want to do anything science related, you'll need math. Physics relies on very advanced mathematics. Biology? You need mathematical models. Computer Science? That's all number systems. Etc. Even if you want to study something other than the hard sciences, there's still a very good chance you'll encounter maths. History, psychology, sociology, and many others use statistical, mathematical methods to process data. Bottom line: if you want higher education, you're very likely to need maths. Certainly not everything you'll learn in high school, but at least some of it.

- Mathematics is all around us. To truly understand the world, we need to understand mathematics. Without mathematics, we would still be cavemen; we would not be able to build houses without having them fall down. We would certainly not have electricity, let alone computers.

- In modern society it is as important to know at least basic mathematics as it is to know how to read and write. If you don't know anything about maths, you would be very bad at handling money for example.

- Even if you really don't need maths itself (or, more probable, some part of maths) studying it helps you build your analytical skills - which will certainly come in handy in any job and life in general.

For (even) more answers and examples check out http://mathforum.org/dr.math/faq/faq.why.math.html and http://www.mathguide.com/issues/whymath.html.
i am still very confused about this topic. why would you go straight to adding 6 to cancel out the negative 6. shouldn't you take away positive six from the twenty to cancel out the negative six?
That's one way of looking at it. But it may not that obvious that what Sal did and what you're suggesting will yield exactly the same result.
For example
Sal did something like this
6y-6+6 = 6y
And you're suggesting that it should be
6y-6-(-6) = 6y
Notice that if you multiply the negative sign - with -6 it will yield +6 (since negative x negative= positive)
That's why:
6y-6-(-6) = 6y-6+6= 6y
Where did he learn all of this
He paid attention in school, did his work, and studied very hard.
1 Comment
20-7y=6y-6 will equal 13=y right
I understand this
How do you know which term to eliminate first, e.g 20 first or 7y.
It doesn't matter. Either way will give you the same answer:
20-7y = 6y -6
Eliminate 20 first:
20 -20 -7y = 6y - 6 -20
-7y = 6y -26
-7y - 6y = 6y - 6y -26
-13y = -26
-13y/(-13) = -26/(-13)
y = 2
Eliminate 7y first:
20-7y = 6y -6
20 -7y + 7y = 6y +7y -6
20 = 13y -6
20 +6 = 13y - 6 + 6
26 = 13y
26/13 = 13y/13
y = 2
1 Comment
What is the steps to solve equation that have variable on both sides
add ten to both sides: 3x+30=8x
minus 3x from both sides:30=5x
divide both sides by 5: 6=x or x=6

Hope this helps
This is the formula for triangle
1 Comment
20-7y=6y-6 Step 1: +7y =+7y Simplify 20=13y-6 Step 2: +6=+6
Simplify: 26=13y Step 3: 26/13=13y/13 Simplify and you're done! y=2
IS there another way to do this that was not in this video?
yes.....its called transposition
it means that you can take term(6y is a term, so is -6) and move it to the other side of the equation., but you have to do the opposite operation with it.
eg: 20-7y=6y-6
20-7y-6y=-6
20-13y=-6
-13y=-6-20
-13y=-26
y=-26/-13
y=2
hope that helped
At :54, wouldn't it be 20y, because you get rid of the seven and you're left with y?
No, it would just be 20. The 20 didn't have a y to begin with, did it? if it doesn't start out with a variable it won't acquire one later in the problem.
i did not understand , n i have test next week
Just remember to keep the equation balanced. if you take away something from one side, take away from the other side too.
if you need more help just click on "comment" and ask.
Solve equation 5m + 9 = 3(m - 5) + 7
5m+9=3m-15+7
2m+9=-8
2m=-17
m=-17/2
Is there a certain side I should put the variable on? like which side would make solving the problem easier for me? (reference, 0:33)
Chole,
You can isolate the variable on either side.
Just choose the side that looks easiest to you.
Ifyour final answer is 17 = x you can always reverse that to x = 17. They both mean the same thing. Equality is commutative.

I hope that helps.
Help! I still don't understand it, I tried watching the video five times already but still don't understand!!
If you know how to solve equations with variables on one side, then you know that the purpose of doing this is to solve for the variable by isolating it on one side of the equation. The same is true for equations with variables on both sides.

*HOW TO SOLVE EQUATIONS WITH VARIABLES ON BOTH SIDES:*
_Use SADMEP ( or reverse PEMDAS): subtraction/addition, division/multiplication, exponents/roots, and then parentheses._

Example:
3x + 8 + x = 12x + 84

1) *Combine Like Terms:*
Since 3x and x are on the same** side of the equation and they are like terms, you can simplify them.
3x + x = 4x, so the equation becomes:
4x + 8 = 12x + 84

You need to make it so that there is only a variable on one side of the equation. To do this, you can subtract 4x from both sides of the equation.
4x - 4x = 0
12x - 4x = 8x, so the equation becomes:
8 = 8x + 84

Since there is still a number on the side of the equation with the variable, you need to do more subtraction to get rid of it. You have to subtract 84 from both sides.
84 - 84 = 0
8 - 84 = -76, so the equation becomes:
-76 = 8x

4) *Division/Multiplication:*
You need to make the 8x an x. Since 8 is being multiplied by x, you can do the opposite if multiplication (division) to get rid of it. You have to divide both sides by 8. 8x/8 = x
-76/8 =-9 1/2, so the solution is:
x = 9 1/2

I hope this helps!
Why is it better for all the Variables to be on the right hand side instead of the left hand side?
would you like x=4y+2 or 4y+2=x
When it comes to actually solving the problem, it makes no difference at all. In the end 5=x and x=5 are equivalent statements. It's all based on your preference.
However, many people find it more convenient to put most of the stuff on the right side, with just a single variable on the left. This may be because in most countries around the world, people write from left to right. By putting the larger mess on the right side, you can simplify and equation without having to change a line every time. For example,
x=3y+8(y-5)+21-9y+6(4y-3)+82 = 3y+8y-40+21-9y+24y-18+82 =26y+45
why didnt you just leave the equation at 5/3
u would have 2 have a a mixed number 4 that 1 2/3
So, let's say your problem has parentheses in it. For example:
8(3a+6)=9(2a-4)

I understand you want to isolate the variable, but if you try to divide 9(2a-4) by 8, I don't quite understand how that would work. Please help!
I think that dividing will make it a bit more complex than it needs to be, so instead let's distribute what's outside of the parentheses.
8(3a+6)=9(2a-4)
Distribute the 8 across the parentheses.
8(3a)+8(6)=9(2a-4)
8(3a)=24a, 8(6)=48.
24a+48=9(2a-4)
Distribute the 9 across the parentheses.
24a+48=9(2a)-9(4)
9(2a)=18a, 9(4)=36.
24a+48=18a+36
Now this problem is a bit more like the one shown in the video.
Subtract each side of the equation by 18a+48.
6a=-12
Divide each side of the equation by 6.
a=-2
We have our solution!

I hope this example helps you solve linear equations with parentheses in them!
So you only take away the number that is needed to isolate the variable? I always thought that you take away the number that is next to the variable.
The goal is for the variable to be alone on one side. You want to remove all the other numbers by bringing them to the other side. It is often easiest to do this by starting with simpler functions (- or +).
Does the variable have to be on both sides?
In most problems you will have 2 variables but in some problems with multiple variables you will some on one side and some on the other side.
y the video r not working
The variables on both side videos don't seem to be working and that's what I need the most help with. I'll continue to look for more and thank you for this site.
Why do you have to drop down the negative during the equation. But then when you divided -13y=-26 you got a positive answer?
u coulda add 6 to both sides to get rid of 6 and then u can add 7y
the asnwer still the same
A choir is singing at a a festival. On the first night, 12 choir members were absent, so the choir stood in 5 equal rows. On the 2nd night, only 1 member was absent, so the choir stood in 6 equal rows. The same number of people stood in each row each night. How many members are in the choir??
Let the choir have x members
the equation will be: (x-12)/5=(x-1)/6 because it said the same number of people stood in each row each night.
6(x-12)=5(x-1)
6x-72=5x-5
x=67
There are 67 members in the choir.
b - 1 - 2b = -4 + 4b how would I do this?
b-1-2b=-4+4b.
Combine the b's:
-1-b=-4+4b.
Move the b's to one side and the numbers to the other:
-1+4=4b+b.
Combine the b's and the numbers:
3=5b.
Divide both sides by 5:
3/5=b.

Hope this helps! :)
Can someone walk me through the steps of Sal's example?
For getting the y's on one side, I was taught to take away the smallest number of y's. Here, you would take away -7y from 6y, which leaves you with 13y. If the equation was 10+2y=-3y+7, you would put all the y's on the left hand side. 2y take away -3y which leaves you with 5y. The equation is now 10+5y=7. Minus the 10 from both sides to get rid of it, and you get 5y=-3. Divide -3 by 5, and you get y=-3/5
Is it possible if you make at least one more vid of these kind of problems, I get it (some what ) but I think one more vid of this with different problems during the vid would do the trick.
is there any other examples that dont have the equation next to the equal sign?