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Equations with parentheses

Sal solves the equation -9 - (9x - 6) = 3(4x + 6) using the distributive property. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user joe.dalessio47
    4r−3=3(3r+4)
    I am older than most trying
    To learn algebra II
    To help my grandkids,
    H
    Very old school ...
    Is there a way to find
    Out the value of the letters
    In various problems
    For example the above
    Problem
    What value is the r
    (88 votes)
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    • mr pink green style avatar for user David Severin
      There is an order to solving and you/grandkids should have done the parts at least separately.
      First, distribute when you have a parentheses, 3(3r+4) = 9r + 12, so you have 4r - 3 = 9r + 12
      Second, move the rs to one side by opposites, since 4r is smaller, it is best to subtract 4r from both sides to get -3 = 5r + 12
      Then it is a two step problem, subtract 12 to get -15 = 5r, finally divide by 5 to get r = -3.
      If you want to test your solution, put in -3 where you see an r
      4(-3) - 3 = 3(3(-3)+4)
      -12 -3 = 3(-9 + 4)
      -15 = 3(-5)
      -15=-15 so the answer checks out
      (4 votes)
  • old spice man green style avatar for user Pedro
    Hi!

    What if we were dealing with Inequalities? Would having a negative 9 behind the brackets change the direction of the Inequality?

    Here, have a look:

    -9 times [9x - 6] > 3 times [4x + 6] » Would that initial negative 9 have influence on the direction the Inequality assumes? Instead of having ''greater than'', would we have ''lesser than''? Or do we turn a blind eye to that initial negative 9 and only change the direction of the Inequality if, later on, we need to multiply or divide?

    Hope I explained myself properly.

    Cheers!
    (55 votes)
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  • piceratops tree style avatar for user Harpreet Lota
    why cant you do -9-1 before you use the distributive property in the beginning of the equation?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      PEMDAS rules always apply when you are simplifying each expression on the sides of the equation.
      You can't do the work inside the parentheses because you can't combine unlike terms. But, you have to deal with the parentheses. The only way to remove the parentheses is to use the distributive property.

      If you do -9-1 as your 1st step, you are skipping all the PEMDAS rules prior to additionn/subtraction, which can't be done.

      Hope this helps.
      (45 votes)
  • male robot hal style avatar for user Andrew St. Clair
    Is there a way to memorize all of these steps to the linear equations?
    (6 votes)
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  • blobby green style avatar for user wgualtiere
    what if the the variables are the same value
    (15 votes)
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    • leaf green style avatar for user Luke Hirsch
      Variables are always the same value unless they are a different letter. ie -3x+7(x+5)=5x+38 even though the x appears more than once. So if you solve the equation above you would get x=-3. If you plug in -3 to all of the x's the equation simplifies to 23 = 23 and it checks out.
      (2 votes)
  • area 52 purple style avatar for user KNIGHTSTER
    what does parenthesis mean?
    (6 votes)
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  • piceratops sapling style avatar for user Jacquelyn  Caldwell
    How do you know if it is negative and not just subtraction? At point 50 seconds in the video he says negative one times negative six. How would you know it's not just subtraction?
    (11 votes)
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    • starky ultimate style avatar for user Patrick
      If you mean something like -7(-3), I can answer that. There are three ways to indicate multiplication: an x sign, a dot, or a number in parentheses touching another number. If it is not one of those, and is more like -5-6, then it would be subtraction. Hope this helps!
      (0 votes)
  • primosaur tree style avatar for user Vraptor008
    What do I do when I have a problem like 5(x➗5-3)=30. I can't get the right answer. I come up with 60.
    (5 votes)
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    • hopper cool style avatar for user Philip
      Here is a step by step solution to 5(x➗5-3)=30
      1. Divide both sides by 5, which gives *(x➗5-3)=6*
      2. Add both sides by 3: x➗5=9
      3. Multiply both sides by 5: x=45

      P.S. if you are able to list the steps you use in solving the equation, then I might be able to help you in explaining why you obtain 60 as your answer.
      (8 votes)
  • blobby green style avatar for user gavin
    how do i do this 8(10-k)=k
    (5 votes)
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  • blobby green style avatar for user 0070194758
    Hi I am going to 7th grade and going to be learning algebra 1 what are some tips I can use to help me understand this topic more.
    (3 votes)
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    • hopper cool style avatar for user Seed Something
      Great question, and smart of you to prepare!

      Algebra uses and builds upon all past Arithmetic math skills and concepts, and both are used in Geometry.

      It's very useful in life and is applied in many different situations, so we can find the answers we need.

      Algebra is a method to solve for unknown values, using logic, balance, and Arithmetic.

      so…
      Review Arithmetic, (algebra is full of fractions, exponents, signed integers, etc), and…

      Take advantage of, and learn from Khan Academy's Algebra prep classes.
      Pre-Algebra
      Get Ready for Algebra 1

      Continued in Comments
      (8 votes)

Video transcript

We have the equation negative 9 minus this whole expression, 9x minus 6-- this whole thing is being subtracted from negative 9-- is equal to 3 times this whole expression, 4x plus 6. Now, a good place to start is to just get rid of these parentheses. And the best way to get rid of these parentheses is to kind of multiply them out. This has a negative 1-- you just see a minus here, but it's just really the same thing as having a negative 1-- times this quantity. And here you have a 3 times this quantity. So let's multiply it out using the distributive property. So the left-hand side of our equation, we have our negative 9. And then we want to multiply the negative 1 times each of these terms. So negative 1 times 9x is negative 9x, and then negative 1 times negative 6 is plus 6, or positive 6. And then that is going to be equal to-- let's distribute the 3-- 3 times 4x is 12x. And then 3 times 6 is 18. Now what we want to do, let's combine our constant terms, if we can. We have a negative 9 and a 6 here, on this side, we've combined all of our like terms. We can't combine a 12x and an 18, so let's combine this. So let's combine the negative 9 and the 6, our two constant terms on the left-hand side of the equation. So we're going to have this negative 9x. So we're going to have negative 9x plus-- let's see, we have a negative 9 and then plus 6-- so negative 9 plus 6 is negative 3. So we're going to have a negative 9x, and then we have a negative 3, so minus 3 right here. That's the negative 9 plus the 6, and that is equal to 12x plus 18. Now, we want to group all the x terms on one side of the equation, and all of the constant terms-- the negative 3 and the positive 18 on the other side-- I like to always have my x terms on the left-hand side, if I can. You don't have to have them on the left, so let's do that. So if I want all my x terms on the left, I have to get rid of this 12x from the right. And the best way to do that is to subtract 12x from both sides of the equation. So let me subtract 12x from the right, and subtract 12x from the left. Now, on the left-hand side, I have negative 9x minus 12x. So negative 9 minus 12, that's negative 21. Negative 21x minus 3 is equal to-- 12x minus 12x, well, that's just nothing. That's 0. So I could write a 0 here, but I don't have to write anything. That was the whole point of subtracting the 12x from the left-hand side. And that is going to be equal to-- so on the right-hand side, we just are left with an 18. We are just left with that 18 here. These guys canceled out. Now, let's get rid of this negative 3 from the left-hand side. So on the left-hand side, we only have x terms, and on the right-hand side, we only have constant terms. So the best way to cancel out a negative 3 is to add 3. So it cancels out to 0. So we're going to add 3 to the left, let's add 3 to the right. And we get-- the left-hand side of the equation, we have the negative 21x, no other x term to add or subtract here, so we have negative 21x. The negative 3 and the plus 3, or the positive 3, cancel out-- that was the whole point-- equals-- what's 18 plus 3? 18 plus 3 is 21. So now we have negative 21x is equal to 21. And we want to solve for x. So if you have something times x, and you just want it to be an x, let's divide by that something. And in this case, that something is negative 21. So let's divide both sides of this equation by negative 21. Divide both sides by negative 21. The left-hand side, negative 21 divided by negative 21, you're just left with an x. That was the whole point behind dividing by negative 21. And we get x is equal to-- what's 21 divided by negative 21? Well, that's just negative 1. Right? You have the positive version divided by the negative version of itself, so it's just negative 1. So that is our answer. Now let's verify that this actually works for that original equation. So let's substitute negative 1 into that original equation. So we have negative 9-- I'll do it over here; I'll do it in a different color than we've been using-- we have negative 9 minus-- that 1 wasn't there originally, it's there implicitly-- minus 9 times negative 1. 9 times-- I'll put negative 1 in parentheses-- minus 6 is equal to-- well, actually, let me just solve for the left-hand side when I substitute a negative 1 there. So the left-hand side becomes negative 9, minus 9 times negative 1 is negative 9, minus 6. And so this is negative 9 minus-- in parentheses-- negative 9 minus 6 is negative 15. So this is equal to negative 15. And so we get negative 9-- let me make sure I did that-- negative 9 minus 6, yep, negative 15. So negative 9 minus negative 15, that's the same thing as negative 9 plus 15, which is 6. So that's what we get on the left-hand side of the equation when we substitute x is equal to negative 1. We get that it equals 6. So let's see what happens when we substitute negative 1 to the right-hand side of the equation. I'll do it in green. We get 3 times 4 times negative 1 plus 6. So that is 3 times negative 4 plus 6. Negative 4 plus 6 is 2. So it's 3 times 2, which is also 6. So when x is equal to negative 1, you substitute here, the left-hand side becomes 6, and the right-hand side becomes 6. So this definitely works out.