Two-step equations with decimals and fractions
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- Let's get some practice solving equations that involve fractions and decimals. And this equation clearly involves fractions. Let's see. We have negative 1/3 = j/4 - 10/3. So I encourage you to pause the video and see if you could solve for j. What j would make this equation true? Alright. Now let's work through this together. So what I like to do, I like to isolate the variable that I'm trying to solve for on one side. Since it's already on the right-hand side, let's try to get all the things that involve j on the right-hand side, and then get rid of everything else on the right-hand side. So I want to get rid of this negative 10/3, and the best way I can think of doing that is by adding 10/3. Now I can't just do that to one side of the equation. Then it wouldn't be equal anymore. If this is equal to that, in order for the quantity to be true, whatever I do to this I have to do to that as well. So I have to add 10/3 to both sides. I have to add 10/3 to both sides of the equation. And so what am I going to get? On the left-hand side, I'm going to have negative 1/3 + 10/3, which is 9/3. 9/3. And then that's going to be equal to. And on the right-hand side, the negative 10/3 and the positive 10/3, those cancel out to get a zero, and I'm just left with j/4. It's equal to j/4. Now you might recognize 9/3, that's the same thing as nine divided by three. So this is just going to be three. So that simplifies a little bit. Three, let me just rewrite it so you don't get confused. Three is equal to j/4. Now, to solve for j, I could just multiply both sides by four. 'Cause if I divide something by four and then multiply by four, I'm just going to be left with that something. If I start with j and I divide by four, and then I multiply, and then I multiply by four, so I'm just going to multiply by four, then I'm just going to be left with j on the right-hand side. But I can't just multiply the right-hand side by four. I have to do it with the left-hand side as well. So I multiply the left-hand side by four as well. And what I will be left with, four times three is twelve. And then j divided by four times four, well that's just going to be j. So we get j is equal to 12. And the neat thing about equations is you can verify that you indeed got the right answer. You can substitute 12 for j here, and verify that negative 1/3 is equal to 12/4 - 10/3. Does this actually work out? Well 12/4 is the same thing as three, and if I wanted to write that as thirds, this is the same thing as 9/3. And 9/3 - 10/3 is indeed equal to negative 1/3. So we feel very good about that. Let's do another example. So I have n/5 + 0.6 = 2. So let's isolate this term that involves n on the left-hand side. So let's get rid of this 0.6. So let's subtract 0.6 from the left-hand side. But I can't just do it from the left. I have to do it from both sides if I want the equality to hold true. So, subtract 0.6. Now on the left-hand side, I'm just going to be left with n/5, and on the right-hand side, 2 - 0.6, that;'s going to be 1.4. And if you don't want to do this in your head, you could work this out separately. It's going to be 2.0 - 0.6. You could say, "Oh, this is 20/10-6/10" which is going to be 14/10, which is that there. Or if you want to do it a little bit kind of the traditional method, you say, "Oh, I'm trying to subtract six from zero, let me re-group." That's going to be a 10. I'm going to take from the ones place. If I take a one from the one's place, and that's going to be equal to 10/10. 10/10 - 6/10 is 4/10. And then, bring down one one minus zero ones is just one. So it's 1.4. And now, to solve for n. Well on the left have n being divided by five. If I just want n here, I can just multiply by five. So, if I multiply by five, five times n divided by five is going to be just n. But I can't just multiply the left-hand side by five. I have to multiply the right-hand side by five as well. And so what is that going to get us? We are going to get n = 1.4 x 5. 1.4 x 5. Now you might be able to do this in your head, 'cause this is one and 2/5. So this thing should all be equal to seven, but I'll just do it this way as well. Five times four is 20. Re-group the two. One times five is five plus two is seven. And when I look at all the numbers that I'm multiplying, I have one digit to the right of the decimal point. So my answer will have one digit to the right of the decimal point. So it's 7.0, or just 7. n = 7. And you can verify this works, 'cause seven divided by five is going to be equal to 1.4, plus 0.6 is equal to two. Let's do one more example. This is too much fun. Alright. 0.5 times the whole quantity (r + 2.75) = 3. Now there's a bunch of ways that you could tackle this. A lot of times when you see something like this, your temptation might be, "Let's distribute the 0.5." But that makes it a little bit hairy, 'cause 0.5 times 2.75. You could calculate that, and you will get the right answer if you do it correctly. But a simpler thing might be, well, let's just divide both sides by 0.5. That way I'm going to get more whole numbers involved. So if I divide. Remember, whatever I do to the left-hand side I have to do to the right-hand side. And the way my brain thought about it was, well, if I divide by 0.5 on the left-hand side I can get rid of this. And if I divide by 0.5 on the right-hand side I'm still going to get an integer. Three divided by 0.5 is six. It's the same thing as three divided by a half. How many halves fit into three? Six halves fit into three. So this going to be six right over here. So these cancel out. And then this is going to be equal to six. So the whole thing is simplified now to r + 2.75 is going to be equal to is equal to six. And now to just isolate the r on the left-hand side, I can substract the 2.75 from the left. But, like we've seen multiple times, I can't just do it from, I can't just do it from the left. I'm having, my brain is malfunctioning. 2.75. I can't just do it from the left. I have to do it from the right as well. 2.75. So this simplifies to r, this simplifies to r, Is equal to -- what's six minus 2.75? Well if you want to do it in your head, six minus two would be four, and then if you take 0.75 from that it would be 3.25. If you don't feel comfortable doing it in your head, we could just write it out. 6.00 - 2.75, be careful to align the decimals and then I've got to do some re-grouping. Let's see, I have zero hundreds, trying to subtract five hundreds. That's not going to work out. So I try to re-group from here, but I have nothing here. So let me re-group from here. One one. Let me take away one of these ones. I'm going to have five ones. And then that's going to be equivalent, so I have five ones here, and that one one I took away is going to be ten tenths. And then I can take one of those tenths away. So I'm going to have nine tenths. And that's going to be ten hundredths. And now I could subtract. Ten hundreths minus five hundredths is five hundredths. Nine tenths minus seven tenths is two tenths. My decimal is going to be there. Five minus two is three. 3.25. And you can verify that. 3.25 + 2.75 is six, times 0.5 is indeed equal to three. So we feel, once again, really good about this.