 This video will talk about multi-step equations. When you have multi-steps, that means that we're going to have to combine some things on one side or do more than one thing than just adding or subtracting like we've done previously. So we need to simplify each side by either combining light terms or using the distributive property. And then if we have variables on both sides, we need to make sure that we collect them, the one side. And if we have constants on both sides, we need to take them to the other side. And then, of course, we check our solutions. So let's look at our examples here. I have light terms on the left-hand side. So I need to combine those light terms, first of all. So negative 3t plus 4t will be t plus 5 equals 0. Now I have constants on both sides. And I want it on the opposite side of my variable. So I want to take the 5 to the other direction. So I'm going to subtract 5 from both sides. And we have t equal negative 5. And if we check it, negative 3 times negative 5 plus 5 plus 4 times negative 5 should be equal to 0. Negative 3 times negative 5 is positive 15 plus 5. And then we have to multiply. 4 times negative 5 is minus 20. And then order of operation tells us to add these first two. So 20 minus 20 minus is equal to 0. And we know that to be true. So t is equal to negative 5. Let's try this problem over here. 5y minus 1 equal 4y plus 4. I have y's on both sides and constants on both sides. So I have to move them. Now we can move either direction. We can put y on either side. We can put the variable on either side. It doesn't matter. I choose to try to move things so that I can keep my variable positive because I'm just a positive person. So let's say 5y and 4y, well, the 5y is larger than 4y. So I want to subtract 4y from this side so that I can keep my y positive on the left-hand side here. And I'm only moving one thing at a time. So 5y minus 4y is y. And then minus the 1 equal to my 4y is already canceled out plus 4 or positive 4 on that side. Now I need to add 1 to both sides. So we add 1 and we get y, those cancel out, equal to 4 plus 1, which is 5. Let's check. 5 times 5 minus 1 should be equal to 4 times 5 plus 4. 5 times 5 is 25 minus 1. And if I finish this side out, 25 minus 1 is 24. Going to the other side now, 5 times 4 is 20 plus 4. And 20 plus 4 is 24. 24 equal 24, so sure enough, y equal 5. Let's look at this one. Negative 7 over 12x plus 2 is equal to 512 over 12x minus 6. Again, we have both kinds of things on both sides. This is a negative 7, and if I add it to the other side, positive 7 will be bigger than 5. So I'm going to, and it's the same sign, so it's going to keep it positive. So I'm going to add 7 over 12x to both sides. So those cancel out, and that's 2. Remember, if you don't like it, you can say 5 over 12 plus 7 over 12, do it sideways. And 5 plus 7 is equal to 12 over 12, so it's really just 1x minus 6. Now I have this 6 on the same side with the x, and I don't want it. I need it to go to the other side, so I'm going to add 6 to both sides. 2 plus 6 is 8 and x. So we think that x is equal to 8, so we check it. 7 over 12 times 8 over 1 would be like fractions. Plus 2 is equal to 5 over 12 times 8 over 1 minus 6. Now we multiply. So negative 56 over 12 plus 2 is equal to 40 over 12 minus 6. I've got to make this 2 a fraction. So minus 56 over 12, I have to multiply by 12 over 12, so I can get a denominator of 12 here, because that's just a 2 over 1. So 2 times 12 would be 24. 1 times 12 is 12. Finishing this side up, we get a negative 32. Negative 56 plus 24 is a negative 32 over 12. And on this side, we have 40 over 12. But again, we have to multiply by 12. That's a minus. To get 1 times 12 would be 12 on the bottom, so 12 times 6 is 72. And 40 minus 72 is a negative 32 over 12. I don't need to simplify my fractions. I just wanted to know that they were the same. So x is equal to 8. All right, two more problems. So now this time, we have some distributing that we have to do, because we have a 5 times the quantity on the left-hand side. So we need to distribute the 5 before we can start moving things across. So 5 times x and then plus 5 times 2, which would be 10, is equal to 4 times x minus 4. There was nothing to do on the other side. Now we're ready to start moving things like we've been doing. If I subtract 4 from 5, I'll get a positive 1x. If I go the other way and subtract 5x from 4x, I'll have a negative. So I don't want the negative. I'm going to subtract 4x from both sides. And that will give me, like we said, x plus 10 equal to negative 4. And if I want to get the 10 to the other side, I have to do the opposite. It's adding 10, so I have to subtract 10. So those cancel each other out, and I have x equal to negative 14. And again, of course, we can check. We have to distribute the 2, but we also have to distribute this negative. So 2 times x is 2x. 2 times negative 5 is a negative 10. And then negative 1 times x would be minus x. And negative 1 times 3 would be minus 3 equal to 5. And now I have like terms all over the place, and I have to move things back and forth. So 2x minus x, let's rewrite this. 2x minus x minus 10 minus 3 equal 5. So now we can rewrite this with our like terms together. So 2x minus x will give us x. Negative 10 minus 3 will give us negative 13, because remember it's plus or negative, and that's equal to 5. So we need to add, since we're subtracting 13, we need to add 13 to both sides. And that shows us that x is equal to 18. Now to check it, we put 2 times 18 minus 5. That's equal. Minus the quantity 18 plus 3. And that's supposed to be equal to 5. So order of operation says work inside the parentheses first. So that's 2 times 13 minus inside this parentheses 18 plus 3 will be 21 equal to 5. 2 times 13 will be 26 minus that 21 should be equal to 5. And 26 minus 21 is 5 equal to 5. So we agree that x is equal to 18.