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Writing Equivalent Equations

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Today we're going to talk about writing equivalent equations. Two equations are equivalent to each other if they have the same solution or solution set, even if they look different. So we'll start by doing an example of two equations that are equivalent, and then we'll talk about how you can determine if equations are equivalent or not.

So behind me are two equations that are equivalent. Even though they don't look the same, they have the same solution. So I'm going to solve each of them to find the solution for x, and then I'll plug it back in to show you that that solution works for both equations.

So our first equation is pretty easy. To solve for x, we just need to cancel out the 8 that's being multiplied, so I'm going to divide both sides by 8. These will cancel out, and this will give me that x is equal to negative 4.

So my second equation, because it's equivalent, should have the same solution of x equals to negative 4. So let's see. To solve this, I'm going to start by canceling out my adding operation by subtracting 5 from both sides. These will cancel, and I'll be left with 3x is equal to negative 7 minus 5, which is negative 12. Here I'm going to divide both sides by 3 to cancel out the 3 that's being multiplied by x, and so now I have here also that x is equal to negative 4.

So we see that both of these equations are equivalent because they have the same solution. So let's just double check the solution by plugging it back into our original equation. So if x is equal to negative 4, then if I use my original equation of 8 times x, I'm going to substitute negative 4, and that should be equal to negative 32. 8 times negative 4 is equal to negative 32, so my solution of x equals to negative 4 is correct.

Let's try the same thing for our second equation. So if I plug negative 4 back into my original equation, I should have 3 times negative 4 plus 5 is equal to negative 7. Simplifying this, 3 times negative 4 gives me negative 12. If I add 5 to that, that does give me negative 7, so here again, I see that my solution of negative 4 is true.

So this is an example of two equations that are equivalent. Let's look further at how you can determine if two equations are equivalent. All right, so I've got two equations behind me and I want to see if they're equivalent, so I'm going to solve each of them separately and see if they have the same solution. If you're feeling pretty confident, go ahead and try these on your own, and then check back with us and see if you've got them correct.

So for my first equation, I've got x plus 6 over 7 equals 2. Since 7 is in the denominator of the fraction, that means it's dividing, so to cancel out this dividing by 7, I'm going to multiply by 7 on both sides of my equation. So here, my two 7s will cancel out, and I'll be left with x plus 6, and on the other side, 2 times 7 gives me 14.

Now I'm going to cancel out the adding operation, and I'm going to subtract by 6 on both sides. So here, my plus 6 and subtract 6 will cancel out, and I'll just be left with x, and 14 minus 6 will give me 8. So my solution for my first equation is 8. We know if that these two equations are equivalent, I should also get 8 for my second equation.

All right, so to solve this, I'm going to start by canceling out my subtracting 10 operation by adding 10 to both sides. These will cancel, and I'll be left with just 3x, and on the other side, 8 plus 10 will give me 18. Now I'm going to cancel out my multiplying by 3 by dividing by 3 on both sides. These 3s will cancel out, and I'll be left with just my x variable. 18 divided by 3 equals 6. So we can clearly see that these two equations are not equivalent because I found different solutions-- x equals 8 and x equals 6-- for both equations.

All right, let's see if these two equations are equivalent to each other. Again, if you're feeling confident, try them on your own, and then check back with us when you're done. So again, I'm going to solve both equations and see if we have the same solution for x.

So for this equation, I need to start by simplifying. I can distribute my 3 to both terms on the inside of the parentheses, so 3 times 5 will give me 15, and 3 times x will give me a positive 3x. Bring down the rest of my equation. So now I can continue to simplify by combining my like terms. I have two constants here, so 15 minus 4 is going to give me 11, and bring down the rest of my terms.

Now I'm going to cancel out my 11. Since this is a positive 11, you can think of it as adding 11, so to cancel out or undo the operation of adding 11, I'm going to subtract 11 from both sides. This will cancel out, and I'll be left with 3x is equal to 6. To cancel out the 3 that's being multiplied by the x, I will divide on both sides. So the 3s cancel, and I'm left with x is equal to 6 divided by 3, or 2.

So let's see if this equation is equivalent to our first one by solving it to see if it has the same solution of x equals 2. So here I can also simplify this equation, because I have, again, two like terms. So negative 3x minus 2x will give me negative 5x, and I'm going to bring down the rest of my equation. I'm going to continue to solve by canceling out my subtracting 8 operation, and I'll add 8 to both sides to do that. These will cancel out, and I'm left with negative 5x is equal to negative 18 plus 8, which is negative 10.

Now I'm going to divide both sides by my negative 5 that's being multiplied by the x. These will cancel out, and I'll be left with x is equal to positive 2. So again, because we found that both equations have a solution of x is equal to positive 2, these two equations, even though they look different, are equivalent.

So let's go over our key points from today. Make sure you get these in your notes if you don't already, so you can refer to them later. So we started by talking about the fact that equations are equivalent to each other if they have the same solution or solution set, and then we showed three examples that if you want to determine if equations are equivalent to each other, you just need to solve each equation and then determine if their solutions are the same.

So I hope that these notes and examples helped you understand a little bit more about writing equivalent equations. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.