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Welcome to Algebra I: Properties of Equality

11 minutes

Hi guys. Welcome to Algebra 1. Today's lesson will focus on the properties of equality. We're getting close to solving multi-step linear equations, and you should understand these properties because they allow us to solve those equations. Okay, you ready? Let's go. Okay. The equality property of addition-- or addition property of equality-- says if we have an equation-- in this case, r equals s-- and r and s are both real numbers, then adding t-- another real number-- to both sides won't change the value of the equation. Rather, it will change the value, but it will keep it balanced. If I have r equals s, r plus t is equal to s plus t. Adding the same thing to both sides still keeps the equation equal and balanced. All right? Okay. Subtraction property of equality. Again, r, s, and t are real numbers. We're saying here that if r equals s, then r minus t is equal to s minus t. So in other words, subtracting the same value from both sides of our equation does not change its equality. The equation is still balanced, as long as I subtract the same thing from both sides, okay? Multiplication property of equality. Let's think about this. If the addition property said adding the same thing to both sides didn't change the equality, the subtraction property said subtracting the same thing from both sides doesn't change the equality. Then what do you think the multiplication property means? I'm sure you guessed it. If I multiply the same thing on both sides, it doesn't change the equality. The equation is still balanced. So in other words, if r equals s, then r times t is equal to s times t. As long as I multiply the same thing on both sides, the equality stays intact. It's still balanced. Let's keep going. Division property of equality. Same general idea: r, s, and t are real numbers, and if r equals s, then r divided by t is equal to s divided by t. As long as I'm dividing the same thing on each side, or dividing each side by the same thing, the balance of the equation still stays intact. It's still equal, all right? Okay, let's put this stuff to work here. Let's look at this equation. It looks like a lot-- there's more steps I'll scroll to show you. We'll work our way through here. With these property symbols on the right, we figure out which property allowed us to transition between the two steps. Let's go ahead and get started. Step 1: I have 5 times that quantity, (x plus 2), equals 30. Basically they're giving you the equation right here. Step 2: Now the left side is 5x plus 10 equals 30. Let's throw back to the properties of real numbers. Which property let us move between steps 1 and 2? Right, it's the one you've seen probably since sixth or seventh grade. The distributive property allowed us move between step 1 and 2. We'll focus on the properties of equality. We're going to keep moving. From step 2 to step 3, so now I have 5x plus 10 minus 10 equals 30 minus 10. Then, step 4: I have 5x equals 20. I see the little property on the right. So which property let us move from step 3 to step 4? If we look at exactly what happened, we subtracted 10 on the left side, and we subtracted 10 on the right side. We subtracted the same thing from both sides. Which property of equality said that we can subtract the same real number from both sides without changing the balance of our equation? Do you remember? It was the subtraction property. Right here I'm going to put "subtraction." Good job. Let's keep going. There's a few more steps. Let me scroll a little bit. Six steps here. We were right here at step 4: 5x equals 20. Step 5: 5x equals 20. And they divided each side of the equation by 5. Step 6: We have x equals 4. Let's see what happened. Between step 5 and step 6, what happened? They divided both sides of the equation by the same real number-- in this case, 5. So which property of real numbers said that you can divide both sides of your equation by the same term and everything's fine? It doesn't change the balance of your equation. Do you remember? Division property, right. So right here, let's put "division." That's what allowed us to move between step 5 and step 6. So not too bad. In solving equations in Pre-Algebra, you've done this, without realizing what you're exactly you're doing, maybe, or why you're able to do that. You know these steps, how to move through an equation. We're giving the properties that justify why we're able to make those moves, okay? All right, let's try this one. We'll work our way through all the steps of this equation. When you see the little property box, write down what property allows us to move between those two steps. You've been doing this, solving equations, all along. From Pre-Algebra, you learned what you can do to both sides of the equation to work through it. We're putting those steps on paper. We're writing down exactly what justifies all the moves we make to solve an equation. Let's look here. Step 1. We have negative x minus 4 equals negative 2, and then in parentheses, we have x plus 3. Step 1, they're just giving you the problem there. Step 2: We still have the same thing on the left-- negative x minus 4. But the right side, now it's negative 2x minus 6. This is a throwback here. It's a property of real numbers. Do you remember what allowed us to move between those two steps? Distributive property, right. Good, good, good. Step 3. We have negative x minus 4 plus 2x equals negative 2x plus 2x minus 6. Just take that in. Step 4: x minus 4 equals negative 6. I see the little property box. Let me set this up and you can figure out the property. In black, it looks like they're telling me I added 2x to both sides of my equation. Think about which property of equality tells you that you can add the same real number to both sides and it doesn't change the balance of it? Think about that. Press pause. To check your answer, go ahead and press play again. Okay, let's look. Adding the same thing to both sides doesn't change the balance; that was the addition property. Good--that's the property that justifies the move from step 3 to step 4. All right, let me scroll down a little bit. You've got one more to try for this problem. Let me get you there. Step 4. So we have x minus 4 equals negative 6. Step 5: We have x minus 4 plus 4 equals negative 6 plus 4. It should look a little familiar. Step 6: x equals negative 2. Let's look here. What property says that I can add the same real number to both sides and it doesn't change anything? You got it, it was another repeat. It's also the addition property. All right! Great job, guys. I hope you're feeling comfortable with the properties of equality. You're ready to solve multi-step linear equations. See you soon for that! Bye. Accessibility provided by the U.S. Department of Education.

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In this program, students learn about the properties of equality. These properties allow students to balance and solve equations involving real numbers. Part of "Welcome to Algebra I" series.

Media Details

Runtime: 11 minutes

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