Question Video: Rearranging and Solving Linear Equations with Decimals

1.5𝑥 + 5 = 3𝑥 − 1.

04:26

Video Transcript

We’ve got one point five 𝑥 plus five is equal to three 𝑥 minus one, so the numbers aren’t always as nice as we’d like. But basically we could, looking at the inverse operations to eliminate 𝑥 from one side, we could either subtract one point five 𝑥 from both sides or we could subtract three 𝑥 from both sides. If we subtract one point five 𝑥 from the left-hand side, that will eliminate 𝑥. And if we subtract one point five 𝑥 on the right-hand side, we’ll have three 𝑥 minus one point five 𝑥. We’ll have positive one point five 𝑥 on the right-hand side. So that looks like that’s gonna be a good plan. Let’s just check it the other way around. If we subtract three 𝑥 from the right-hand side, we’ll have no 𝑥s; three 𝑥 minus three 𝑥 is nothing. If we subtract three 𝑥 from the left-hand side, one point five 𝑥 take away three 𝑥 is negative one point five 𝑥, which will be a bad thing, so I’m not going to do that. So subtracting one point five 𝑥 from the left-hand side, we started off with one point five 𝑥 and we’re taking away one point five 𝑥. So we’ve got no 𝑥s. So we’re just left with five. And on the right-hand side, we started off with three 𝑥s. We’re taking away one point five 𝑥s, we’re leaving ourselves with one point five 𝑥 and the negative one. So now I’ve got one point five 𝑥 minus one. Well the inverse operation of taking away one is adding one. So with the interest of just eliminating everything else and leaving the 𝑥 term on its own, I’m gonna add one to both sides of that equation. Then on the right-hand side, I’ve got negative one add one is nothing. So it just leaves me with one point five 𝑥. And on the left-hand side, five plus one is six. So now I’m left with one point five 𝑥 is equal to six. Well the inverse operation to multiplying by one point five, which is what we’re doing here, is dividing by one point five. Now if I’m comfortable dividing both sides by one point five, how many one point five’s go into six? If we can spot that, then I would recommend dividing by one point five at this time. And in fact, I happen to know one point five plus one point five is three. So one point five goes into six four times. So I could do that one in my head. But if you don’t feel comfortable with that, another top tip is to just try to get rid of those decimals at this stage by maybe doubling both sides of our equation. So two, lots of six on the left-hand side are twelve. And two, lots of one point five 𝑥 on the right-hand side make three 𝑥. Now I can do the inverse operation of multiplying by three, which is dividing by three. And that’s a nice whole number; it’s a bit easier to work with. So on the right-hand side, I’ve got three 𝑥 divided by three, which is just one 𝑥. And on the right-hand side, twelve divided by three is four. So our answer is four is equal to 𝑥 or 𝑥 is equal to four. And now if I have time, I can do the left-hand-right-hand check again if I wanted to. But in fact, I’m just gonna to take a moment to show you another slightly shorter way of doing all the working out. And I’ve been writing it out, all the equations again and adding one and subtracting one and-and showing all of that very clearly. But there’s a bit of a shortcut you can take when you’re used to doing that and you know what you’re doing. So let’s have a look at that. So in stage one, we said we were gonna to subtract one point five 𝑥 from both sides of our equation. Now I can recognise that on the left-hand side, I’ve got one point five 𝑥. I’m taking away one point five 𝑥. I don’t really need to write it all out like this. I can just do that straight away and write down the answer five. And then on the right-hand side, three 𝑥 take away one point five 𝑥 is just one point five 𝑥. And I’ve also got minus one. Now I’m gonna to add one to both sides. So that’s the inverse operation of subtracting one, so I can just do that calculation straight away: five plus one is six. So I can just write that down. And on the right-hand side, one point five 𝑥 minus one plus one is just one point five 𝑥. Now I’m going to double both sides like I did before. And again, I can just write those answers down. And now I’m gonna divide both sides by three. And again, I can just write the answers down; twelve divided by three is four; three 𝑥 divided by three is just 𝑥. So you see that we end up with a much more efficient-looking solution with a lot less writing. We’ve still got pretty clear commentary on each stage of the process, but we just have less writing to wade through in order to get to our answer.

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