Solving Equations by Multiplying and Dividing

This lesson will continue our work solving equations by demonstrating how to multiply and divide to isolate a variable. By asking what we are doing to “x” and performing the inverse (opposite) operation, we will see exactly how to solve equations where adding and subtracting are not enough.

So without further ado, read through the slides below to get a feel for how to solve equations using multiplication and division!

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

Rational and Irrational Numbers

This lesson is a video generated from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. You may also head over to my YouTube channel to see the slideshows in video form there!

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

  • Solving equations
  • By multiplying or dividing
  • Objectives
  • By the end of this lesson you should feel comfortable:
  • Isolating variables utilizing the multiplication and division properties of equality.
  • Multiplication property of equality
  • Assuming you watched the last lesson, we should be able to introduce our final two properties without much stress.
  • First, the multiplication property of equality states that we can multiply one side of an equation by any number as long as we multiply the other side by the same number.
  • Thinking back to our scale, if we doubled the amount in one scale, then the other must also be doubled to remain balanced.
  • Notice the notation here where we use a dot, •, to represent multiplication so as not to get confused with our variable, x.
  • Solving equations with multiplication
  • More importantly, how can we use this to solve for x?
  • Well, let’s bring back the problem we just had:
  • In this example, x is being divided by 2, or “a number divided by 2 is 8.”
  • Our three steps should go like this now:
  • Multiply both sides by whatever x is being divided by.
  • Cancel the number being multiplied and divided to get x by itself.
  • Multiply the other side to get your answer.
  • Viola! You’ve solved an equation using multiplication!
  • Division Property of Equality
  • There should be a pattern you’re discovering by now. So without further ado, our last equality property:
  • The division property of equality states that you can divide one side of an equation by any number other than 0 as long as you also divide the other side by the same number.
  • It is very important here to note the difference in this definition from the other equality properties.
  • We can never divide by 0. Any other number is fair game.
  • Solving equations with division
  • Time for one final example:
  • When a number appears next to a variable, this is our third and final way of representing multiplication.
  • To solve, we are all about opposites, or inverse operations, if you’d like to sound fancy.
  • Divide both sides by whatever is being multiplied to x.
  • Cancel the numbers being divided and multiplied to get x by itself.
  • Divide on the other side to arrive at the solution!
  • conclusion
  • The multiplication property of equality states that we can multiply one side of an equation by any number as long as we multiply the other side by the same number.
  • The division property of equality states that you can divide one side of an equation by any number other than 0 as long as you also divide the other side by the same number.
  • We multiply to cancel out division when we see a variable being divided by a number.
  • We use division to cancel out multiplication when we see a variable being multiplied by a number.
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