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One-step multiplication & division equations

Learn to solve equations like "4x = 20" or "y/3 = 7".
Based on our understanding of the balance beam model, we know that to keep a true equation, we always have to do the same thing to both sides of an equation.
But how do we know what to do to both sides of the equation?

Multiplication and division are inverse operations

Here's an example of how division is the inverse operation of multiplication:
If we start with 7, multiply by 3, then divide by 3, we get back to 7:
73÷3=7
Here's an example of how multiplication is the inverse operation of division:
If we start with 8, divide by 4, then multiply by 4, we get back to 8:
8÷44=8

Solving a multiplication equation using inverse operations

Let's think about how we can solve for t in the following equation:
6t=54
We want to get t by itself on the left hand side of the equation. So, what can we do to undo multiplying by 6?
We should divide by 6 because the inverse operation of multiplication is division!
Here's how dividing by 6 on each side looks:
6t=546t6=546          Divide each side by six.t=9          Simplify.

Let's check our work.

It's always a good idea to check our solution in the original equation to make sure we didn't make any mistakes:
6t=5469=?5454=54
Yes, t=9 is a solution!

Solving a division equation using inverse operations

Now, let's try to solve a slightly different type of equation:
x5=7
We want to get x by itself on the left hand side of the equation. So, what can we do to cancel out dividing by 5?
We can multiply by 5 because the inverse operation of division is multiplication!
Here's how multiplying by 5 on each side looks:
x5=7x55=75          Multiply each side by five.x=35          Simplify.

Let's check our work.

x5=7355=?77=7
Yes, x=35 is a solution!

Summary of how to solve multiplication and division equations

Awesome! We just solved a multiplication equation and a division equation. Let's summarize what we did:
Type of equationExampleFirst step
Multiplication equation6t=54Divide each side by six.
Division equationx5=7Multiply each side by five.

Let's try solving equations.

Equation A
Which operation would help solve for w?
8w=72
Choose 1 answer:
w=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Want to join the conversation?

  • duskpin sapling style avatar for user Makayla
    So even if you have a fraction it just means to divide but you should multiply?
    (60 votes)
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  • aqualine ultimate style avatar for user jorge mendoza
    i dont get this, it says "what you do to one side, you do to the other" so if you have x/5=7 you have to multiply each side by 5, so is it 5x5=25 and 7x5=35? if so what happened to the 25?
    (14 votes)
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  • starky seedling style avatar for user Bailey
    the only one I don't understand is the first one,any tips?
    (17 votes)
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  • blobby green style avatar for user samaimoore126
    Is there any way i cant get taught better for inverse operation
    (11 votes)
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    • duskpin ultimate style avatar for user Azza Safa
      If you are looking for another video or some reference to use for inverse operation, I would recommend using Math Antics or Mathceraptops (YouTube channel).This are the places i personally do to for second opinions. I am not sure if Math antics has a video about inverse operation but Mathceraptops sure do.
      (17 votes)
  • duskpin seedling style avatar for user vaca,Natalia
    If a letter is close to a number how do you solve the equation?
    (4 votes)
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    • blobby green style avatar for user Maggie
      I'm assuming your referring to something such as 7x = 14
      in this case, you would use 7x to say that 7 of that variable is equal to 14 in which case x = 2. This works because 7x is used to refer to multiplication easily during algebra since if you tried 7 x X = 14, it can get a bit confusing. While you can also use
      7 ⋅ X = 14 (⋅ being another form of multiplication symbol) but it's much less common.

      TLDR; when a number is by a variable it means to multiply the variable by the number.
      (6 votes)
  • blobby green style avatar for user Beansbeansbeans
    this is too easy.
    (7 votes)
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  • blobby green style avatar for user Hakase
    About the "what you do to one side, you do to the other".
    To understand the logic behind this, first let's imagine a beam balance in front of you, it has two blocks on each side each weighing 200 grams, thus the two sides are equal.

    However if you add another 100g block on one side, the balance becomes uneven, as in 200g < 300g . To solve this we add another 100g block to the other side, so the beam balance becomes even again, as in 300g = 300g . That's why what you do on one side you also do to the other, when solving an equation.

    TL:DR; We do that to avoid a false equation.
    (8 votes)
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  • blobby green style avatar for user TylerG
    I am your question.
    (8 votes)
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  • blobby green style avatar for user KATHERINE S
    It is the same question so will the answer change or stay the same?
    (6 votes)
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  • winston default style avatar for user Victoria
    I don't get any of this. ... I tried to understand it but I can't.
    ;-; Can someone help me please? I am confused how u get a fraction number when ur doing this. Help please :/
    (7 votes)
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    • blobby green style avatar for user TechGuru2000
      Ok, let's start simply, with this equation:w5=55, which can also be written as w.5=55.

      The dot represents the multiplication sign, and if the letter and the number are joined together, that also represents multiplication.

      In this Lesson, we were taught that to solve for w in w.5=55, we need to use the inverse operation, which basically means the opposite of the operations, addition is the opposite of subtraction, and multiplication is the opposite of division.

      We were also taught that we need to use the inverse for both places, not just one, so it can be a balanced equation, an example of that is w.5=55 if the . represents multiplication, we divide instead.

      So we would do 55/5, which is the same as a fraction (a fraction is another form of showing division, the / represents division and fractions, and the numerator is divided by the denominator), and 55/5 or 55 divided by 5 is equal to 11, so w is equal to 11, or w=11.


      Now for division, for the equation t/6=7, or 7=t/6 (they are the same thing, just put in a different format, keep that in mind), we use inverse operation, as shown in the Lesson, and we know what type of inverse operation it is because t/6 is division -because it's a fraction-, so we use multiplication instead so to solve for t in t/6=7, we multiply 7 by 6, which, keep in mind, can be represented as 7 . 6, or 7 X 6, and 7 X 6 is equal to 42, so t=42
      (0 votes)