WEBVTT 00:00:00.820 --> 00:00:05.488 In this session we're going to be having a look at simple equations 00:00:05.488 --> 00:00:09.378 in one variable and the equations will be linear. 00:00:09.378 --> 00:00:11.712 That means that there are no x-squared terms 00:00:11.712 --> 00:00:18.371 and no x-cube terms, just x's and numbers. So let's have a 00:00:18.371 --> 00:00:25.286 look at the first one. 3x plus 15 equals x plus 25. 00:00:25.286 --> 00:00:29.896 An important thing to remember about any equation is this equal 00:00:29.896 --> 00:00:34.967 sign represents a balance. What that equal sign says is that 00:00:34.967 --> 00:00:40.960 what's on the left hand side is exactly the same as what's on 00:00:40.960 --> 00:00:43.265 the right hand side. 00:00:43.310 --> 00:00:48.680 If we do anything to one side of the equation, we have to do it 00:00:48.680 --> 00:00:52.618 to the other side. If we don't, the balance is disturbed. 00:00:53.370 --> 00:00:55.106 If we can keep that in mind, 00:00:55.870 --> 00:00:59.740 then whatever operations we perform on either side of the 00:00:59.740 --> 00:01:04.771 equation, so long as it's done in exactly the same way on other side, 00:01:04.771 --> 00:01:06.706 we should be alright. 00:01:07.520 --> 00:01:12.248 Our first step in solving any equation is to attempt to gather 00:01:12.248 --> 00:01:16.188 all the x terms together and all these free floating numbers together. 00:01:16.188 --> 00:01:22.098 To begin with we've got 3x on the left and and x on the right. 00:01:22.098 --> 00:01:28.402 If we take an x away from both sides, we take one x off the left, 00:01:28.402 --> 00:01:34.312 and one x off the right. That will give us 2x plus 15 equals 25. 00:01:34.312 --> 00:01:38.252 We need to get the numbers together. These free numbers. 00:01:38.252 --> 00:01:42.960 We can see that if we take 15 away from the left, and 15 away from the right 00:01:42.960 --> 00:01:47.580 then we will have no numbers remaining on the left, just a 2x 00:01:47.580 --> 00:01:51.870 and taking 15 away on the right, that gives us 10. 00:01:51.870 --> 00:01:53.850 So, x must be equal to 5. 00:01:54.630 --> 00:01:57.479 That was relatively straightforward. 00:01:57.479 --> 00:02:01.956 Lots of plus signs, no minus signs, which we know can be complicated, 00:02:01.956 --> 00:02:05.619 also no brackets. So let's introduce both of those. 00:02:06.650 --> 00:02:14.190 We will take 2x plus 3 is equal to 00:02:14.190 --> 00:02:17.206 6 minus bracket 2x minus three bracket. 00:02:18.410 --> 00:02:23.630 Now this right hand side needs a little bit of dealing with. 00:02:23.630 --> 00:02:28.850 It needs getting into shape so we have to remove 00:02:28.850 --> 00:02:34.940 this bracket. 2x plus 3 equals 6, 00:02:34.940 --> 00:02:39.725 now we take away 2x, thats minus 2x and then we're taking away minus 3. 00:02:39.725 --> 00:02:44.510 When we take away minus 3, that makes it plus 3. 00:02:45.080 --> 00:02:49.724 Before we go any further, we need to tidy this right hand 00:02:49.724 --> 00:02:55.916 side up a little bit more 2x plus 3 is equal to 9 minus 2x 00:02:55.916 --> 00:03:00.947 and now we're in the same position at this line as we were 00:03:00.947 --> 00:03:05.591 when we started there. So we need to get the xs together, 00:03:05.591 --> 00:03:11.396 which we can best do by adding 2x to each side. 00:03:11.396 --> 00:03:15.266 On the right, minus 2x plus 2x. This side gives us no x. 00:03:15.290 --> 00:03:22.235 And on the left it's going to give us 4X. 00:03:22.235 --> 00:03:29.180 4X plus 3 is equal to 9 because minus 2X plus 2X gives no x. 00:03:29.180 --> 00:03:35.199 Now I can take 3 away from each side, 4X equals 6 00:03:35.199 --> 00:03:42.144 and so x is 6 divided by 4 and we want to write that in its 00:03:42.144 --> 00:03:46.774 lowest terms which is 3 over 2. Perfectly acceptable answer. 00:03:46.780 --> 00:03:51.448 Or one and a half. So either of these are acceptable answers. 00:03:51.448 --> 00:03:56.116 This one (6 over 4) isn't because it's not in its lowest terms, it must be reduced 00:03:56.116 --> 00:04:01.173 to its lowest terms. So either of those two are OK as answers. 00:04:01.750 --> 00:04:06.040 With all of these equations, we should strictly check back by 00:04:06.040 --> 00:04:10.720 taking our answer and putting it into the first line of the 00:04:10.720 --> 00:04:15.790 equation and seeing if we get the right answer. 00:04:15.790 --> 00:04:22.420 So with this five, 3 fives are 15, and 15 is 30. 5 plus 25 is also 30 00:04:22.420 --> 00:04:26.710 The balance that we talked about at the beginning is maintained. 00:04:26.710 --> 00:04:31.780 If you take this number, 3 over 2, or one and a half. 00:04:31.800 --> 00:04:35.408 And substitute it back into here we should find again that 00:04:35.408 --> 00:04:39.344 it gives us the right answer, that both sides give the same value. 00:04:39.344 --> 00:04:42.952 They balance. Solet's just do that. 00:04:42.952 --> 00:04:47.544 Let us take one and a half. So 2 times one and a half is 3, 00:04:48.130 --> 00:04:53.520 plus three is 6, so we've got six on the left side. 00:04:53.520 --> 00:04:57.370 Here we have 6, takeaway, now let's deal with this bracket, 00:04:57.370 --> 00:05:01.605 just this bracket on its own, nothing else. 2x takeaway, three, that's nothing, 00:05:01.605 --> 00:05:06.225 so we are just left with 6. We calculated this side to be 6. 00:05:06.225 --> 00:05:11.230 Six equals six, so again, we've got that 00:05:11.230 --> 00:05:15.080 balance, so we know that we've got the right answer. 00:05:16.250 --> 00:05:21.437 Let's carry on and have a look at one or two questions with 00:05:21.437 --> 00:05:25.028 more brackets and this time numbers outside those 00:05:25.028 --> 00:05:30.215 brackets as well. So in a sense, what we're doing is we are 00:05:30.215 --> 00:05:33.806 increasing the complexity of the equation, but the simple 00:05:33.806 --> 00:05:38.993 principles that we've got so far are going to help us out because 00:05:38.993 --> 00:05:42.983 no matter how complicated it gets and this does look 00:05:42.983 --> 00:05:46.574 complicated, the same ideas work all the time. 00:05:46.600 --> 00:05:52.320 line:1 We begin by multiplying out the brackets and taking care, 00:05:52.320 --> 00:05:58.560 line:1 in particular, with any minus signs that come up so 00:05:58.560 --> 00:06:05.320 line:1 8 times by x is 8x and eight times by minus three is minus 24. 00:06:05.320 --> 00:06:09.480 line:1 We've multiplied everything inside that bracket by what's 00:06:09.480 --> 00:06:16.240 line:1 outside, so we've 8 times by x and we've 8 times by minus 3 00:06:16.240 --> 00:06:20.730 line:1 Now we remove this bracket were taking away 6, 00:06:20.730 --> 00:06:24.276 line:1 that's minus six and we're taking away minus 2x, 00:06:24.920 --> 00:06:30.822 line:1 a minus minus gives us a plus. So that's plus two X 00:06:30.822 --> 00:06:38.530 equals, two times x here, is 2x, 2 times by 2 is 4. 00:06:38.530 --> 00:06:44.950 and now we have to multiply by minus five, so we've got 00:06:44.950 --> 00:06:52.440 minus five times by 5. That's minus 25. And minus five times by minus x. 00:06:52.440 --> 00:06:54.580 So that's plus 5x. 00:06:55.300 --> 00:07:01.007 Each side needs tidying up. We need to look at this side and 00:07:01.007 --> 00:07:06.714 gather x terms and numbers together. so with 8x plus 2x that's 10x. 00:07:07.370 --> 00:07:13.717 Take away 24, takeaway six, that's taking away 30 altogether. 00:07:13.717 --> 00:07:16.602 2x and 5x gives us 7x. NOTE Paragraph 00:07:17.370 --> 00:07:22.694 Add on 4 takeaway 25. Now that's the equivalent of taking 00:07:22.694 --> 00:07:29.545 away 21. Let's get the xs together. We can take 7x away from each side 00:07:29.545 --> 00:07:35.330 so we would have 3x minus 30 there equals just 00:07:35.330 --> 00:07:38.000 minus 21 because we've taken that 7x away. 00:07:38.000 --> 00:07:44.314 Add the 30 to each side, because minus 30 + 30 gives us 00:07:44.314 --> 00:07:50.816 nothing. And add it on over here. So we get three x equals, 00:07:50.816 --> 00:07:57.298 30 added to minus 21, just the same as 30 takeaway 21, 00:07:57.298 --> 00:07:59.150 the answer is 9. 00:07:59.170 --> 00:08:03.295 And three times by something to give us 9 means the 00:08:03.295 --> 00:08:08.170 something, the x, must be 3. Again we ought to be able 00:08:08.170 --> 00:08:12.670 to take that 3, put it back into this line and see that 00:08:12.670 --> 00:08:17.545 in fact we've got the correct answer. So let's just try. 00:08:17.545 --> 00:08:22.420 3 minus 3 is zero. 8 times by 0 is still 0, 00:08:22.420 --> 00:08:23.545 so we can forget about that term. 00:08:25.090 --> 00:08:29.640 2 times by 3three is 6, so six takeaway six is nothing, so 00:08:29.640 --> 00:08:33.140 we've actually got nothing there, zero, and nothing there, zero. 00:08:33.140 --> 00:08:37.340 so there's nothing on the left side of the equation. 00:08:37.340 --> 00:08:42.590 The right hand side should come out to 0 as well. Let's see that it does. 00:08:42.590 --> 00:08:48.190 3 plus 2 is 5 and 2 fives are 10. 5 takeaway 3 is 2, 00:08:48.190 --> 00:08:53.090 and 5 twos are 10. So we've got 10 takeaway 10, 00:08:53.090 --> 00:08:55.190 this gives us nothing again, so we've got nothing on the right. 00:08:55.190 --> 00:09:00.334 The equation balances with this value, 00:09:00.334 --> 00:09:03.082 so this is our one and only solution. 00:09:03.082 --> 00:09:10.300 We take a final example of this kind with 00:09:10.300 --> 00:09:15.820 x plus 1 times by 2x plus 1. 00:09:16.370 --> 00:09:23.954 Equals x plus 3 times by 2x plus 3 00:09:23.954 --> 00:09:26.798 - 14. 00:09:27.590 --> 00:09:31.869 Now, I did say that these were linear equations that there 00:09:31.869 --> 00:09:34.592 would be no x-squared terms in them. 00:09:34.592 --> 00:09:38.322 But when we start to multiply out these 00:09:38.322 --> 00:09:41.612 brackets, we are going to get some x-squared terms. 00:09:42.990 --> 00:09:49.150 Let's have a look, to show you what actually does happen. 00:09:49.150 --> 00:09:52.230 We do x times by two x. That's two x-squared. 00:09:52.230 --> 00:09:57.880 x times by one, that is plus x. 00:09:58.530 --> 00:10:01.785 One times two x, that is plus 2x. NOTE Paragraph 00:10:01.785 --> 00:10:05.715 1 times by 1 00:10:05.715 --> 00:10:08.810 + 1. Equals... 00:10:10.110 --> 00:10:13.926 x times by two x, that's two x-squared. 00:10:15.790 --> 00:10:19.500 x times by three is 3x.. 00:10:20.190 --> 00:10:23.640 3 times by 2x is 6x. 00:10:24.490 --> 00:10:30.001 3 times by 3 is 9 and finally takeaway 14. 00:10:30.001 --> 00:10:35.011 Now we need to tidy up both sides here. 00:10:35.870 --> 00:10:42.710 Two x-squared plus 3x plus one equals 00:10:42.710 --> 00:10:48.182 2 x-squared plus 9x, (taking these two terms together) NOTE Paragraph 00:10:48.820 --> 00:10:52.448 Plus 9, minus 14. 00:10:54.510 --> 00:10:58.709 And this needs a little bit more tidying, but before we do that, 00:10:58.709 --> 00:11:03.231 let's just have a look at what we've got. 00:11:03.231 --> 00:11:06.784 We've got 2 x-squared there, and 2 x-squared there. They are both 00:11:06.784 --> 00:11:10.660 positive, so I can take two x-sqaured away from both sides. 00:11:10.660 --> 00:11:14.213 That means that the two x-squared vanishes from both 00:11:14.213 --> 00:11:18.089 sides, so let's do that. Take two x-squared from each side, 00:11:18.089 --> 00:11:22.934 and that leaves us 3x plus one, is equal to 9x, and now we can 00:11:22.934 --> 00:11:24.872 do this bit 9 takeaway 14, 00:11:24.940 --> 00:11:29.417 Well, that's going to be takeaway five. I still have the 00:11:29.417 --> 00:11:34.708 five to subtract. Now we're back to where we used to be getting 00:11:34.708 --> 00:11:38.371 the xs together, getting the numbers together. We can 00:11:38.371 --> 00:11:43.662 subtract 3x from both sides, so that gives me one equals 6x minus 5 00:11:43.662 --> 00:11:49.360 We can add the five to both sides so that NOTE Paragraph 00:11:49.360 --> 00:11:52.616 6 equals 6X, and so 1 is equal to x. 00:11:53.810 --> 00:11:58.118 And again, we can check that this works. We can substitute it back in. 00:11:58.118 --> 00:12:03.862 1 plus 1 is 2, two ones are two, and one is 3. 00:12:03.862 --> 00:12:08.529 So effectively we've got 3 times by two at this side, which gives 6. 00:12:08.529 --> 00:12:16.045 Here 1 + 3 is 4 two times by one is 2, + 3 is 5 00:12:16.045 --> 00:12:21.388 so we've got four times by 5. So altogether there there's 20. 00:12:22.100 --> 00:12:26.940 Takeaway 14 is again 6, so again, this equation is balanced 00:12:26.940 --> 00:12:31.340 exactly when we take x to be equal to 1. 00:12:32.410 --> 00:12:36.700 Now, those equations that we've just looked at have really been 00:12:36.700 --> 00:12:39.820 about whole numbers. The coefficients have been whole 00:12:39.820 --> 00:12:43.330 numbers. Everything's been in terms of integers. What happens 00:12:43.330 --> 00:12:47.230 when we start to get some fractions in there? Some 00:12:47.230 --> 00:12:51.910 rational numbers? How do we deal with that? So again, we're going 00:12:51.910 --> 00:12:56.590 to add in more complexity, but again, the rules are the same. 00:12:56.590 --> 00:13:02.440 What we do to one side of the equation we must do to the other 00:13:02.440 --> 00:13:04.780 side in order to preserve that 00:13:04.780 --> 00:13:10.930 particular balance. So let's introduce some fractions 00:13:10.930 --> 00:13:14.068 along with some brackets. 00:13:14.068 --> 00:13:22.050 4 bracket X plus 2, all over 5 is equal to 00:13:22.050 --> 00:13:25.470 7 plus 5x over 13. 00:13:26.180 --> 00:13:28.950 We've got some fractions here. 00:13:29.790 --> 00:13:35.404 Numbers in the denominator 5 and 30. We want rid of those we want 00:13:35.404 --> 00:13:40.216 to be able to work with whole numbers with integers, so we 00:13:40.216 --> 00:13:45.830 have to find a way of getting rid of them. That means we 00:13:45.830 --> 00:13:50.241 have to multiply everything because what we do to one side, 00:13:50.241 --> 00:13:55.053 we must do to the other. 00:13:55.053 --> 00:13:59.865 The common denominator for five and 13 is 65, that's five times by 13. 00:13:59.865 --> 00:14:05.080 So let's do that. Let's multiply everything by 65 and 00:14:05.080 --> 00:14:10.778 I'll write it down in full so that we can see it happening. 00:14:10.778 --> 00:14:17.290 we have 65 times 4, brackets x+2 over 5. Then we have 65 times by 7. 00:14:17.290 --> 00:14:22.174 Remember, I said we have to multiply everything, so it's not 00:14:22.174 --> 00:14:26.651 just these fraction bits, it's any spare numbers that there are 00:14:26.651 --> 00:14:27.872 around as well. 00:14:28.390 --> 00:14:34.550 Plus 65 times by 5x over 30. 00:14:35.260 --> 00:14:41.070 Now let's look at each term and make it simpler. Tidy it up. 00:14:41.070 --> 00:14:46.880 For a start, five will divide into 65, so 5 into five goes one 00:14:46.880 --> 00:14:49.370 and five into 65 goes 13 times. 00:14:50.110 --> 00:14:57.950 And then four times by 13? Well, that's 52, so we have 00:14:57.950 --> 00:15:01.870 52 times by x +2 equals. 00:15:02.790 --> 00:15:07.422 That's in a nice familiar form where used to that sort of 00:15:07.422 --> 00:15:11.282 former. We've arrived at it by choosing to multiply everything 00:15:11.282 --> 00:15:14.756 by this common denominator. Let's tidy this side up. 00:15:15.490 --> 00:15:21.250 7 times by 65. Well, that's pretty tall order. Let's try it. 00:15:21.250 --> 00:15:26.050 Seven 5s are 35. Seven 6s are 42 and three is 45. 00:15:26.760 --> 00:15:33.956 Here with 65 times 5 x over 13. 13 goes into 13 once, 00:15:33.956 --> 00:15:41.152 and 13 goes into 65 five times. So we five times by 5x is 25x. 00:15:41.152 --> 00:15:46.334 You may say 'what happened to these ones?'. Well if I divide by 00:15:46.334 --> 00:15:49.958 one it stays unchanged so I don't have to write them down. 00:15:50.970 --> 00:15:56.885 And now we left with an equation that were used to 00:15:56.885 --> 00:16:01.435 handling. We've met these kind before, so let's multiply out 00:16:01.435 --> 00:16:05.985 the brackets, get the xs together and solve the equation. 00:16:05.985 --> 00:16:12.355 So we multiply out the brackets 52 times 2 is 104 00:16:12.355 --> 00:16:19.180 equals 455 plus 25x. Take the 25x away from each side, 00:16:19.180 --> 00:16:21.455 gives me 27x there. 00:16:21.470 --> 00:16:27.509 And no x is there. Take the 104 away 00:16:27.509 --> 00:16:32.206 from each side, which gives me 351. 00:16:33.410 --> 00:16:38.701 And now lots of big numbers, really, that shouldn't be a NOTE Paragraph 00:16:38.701 --> 00:16:43.511 problem to us. 351 divided by 27 will certainly go once 00:16:44.190 --> 00:16:48.782 (there's one 27 in the 35 and eight over and 27s into 8 00:16:48.782 --> 00:16:53.374 go three times, so answer is x equals 13 and we 00:16:53.374 --> 00:16:57.310 should go back and check it and make sure that it's right. 00:16:57.310 --> 00:16:58.622 So let's do that. 00:16:59.460 --> 00:17:02.815 13 + 2? Well, that's 15 00:17:02.815 --> 00:17:09.818 15 divided 5 is 3 and now times by 4, so that's 12. 00:17:09.818 --> 00:17:15.306 Hang on to that number 12 at that side. 00:17:15.306 --> 00:17:20.402 5 times 13 divided by 13. So the answer is just five 00:17:20.402 --> 00:17:25.890 plus the 7 is again 12. The same as this side. 00:17:25.890 --> 00:17:29.418 So the answer is correct. It balances. 00:17:30.190 --> 00:17:35.151 Let's practice that one again and have a look at another example. 00:17:35.151 --> 00:17:39.086 We take X plus 5 over 6 00:17:39.086 --> 00:17:45.938 minus x plus one over 9 00:17:46.440 --> 00:17:50.262 is equal to x plus three over 4. 00:17:50.262 --> 00:17:53.842 We haven't got any brackets. 00:17:53.842 --> 00:17:58.077 Does that make any difference? The thing you have 00:17:58.077 --> 00:18:00.075 to remember is that this line. 00:18:00.930 --> 00:18:04.700 Not only acts as a division sign, but it acts as a bracket. 00:18:05.230 --> 00:18:11.665 It means that all of x plus 5 is divided by 6. 00:18:12.410 --> 00:18:18.080 So it might be as well if we kept that in mind and put 00:18:18.080 --> 00:18:22.535 brackets around these terms so that we're clear that 00:18:22.535 --> 00:18:27.395 we've written down that these are to be kept together and are 00:18:27.395 --> 00:18:32.660 all divided by 6. These two are to be kept together and all 00:18:32.660 --> 00:18:37.115 divided by 9. And similarly here they are all divided by 4. 00:18:37.970 --> 00:18:42.799 Next step we need a common denominator. We need a number 00:18:42.799 --> 00:18:47.628 into which all of these will divide exactly. Now we could 00:18:47.628 --> 00:18:50.701 multiply them altogether and we'd be certain. 00:18:51.250 --> 00:18:57.581 But the arithmetic would be horrendous. 6 times 9 times 4 is very big. 00:18:57.581 --> 00:19:03.425 Can we find a smaller number into which six, nine, and 00:19:03.425 --> 00:19:05.373 four will all divide? 00:19:06.010 --> 00:19:12.406 Well, a candidate for that is 36. 36 will divide by 6l 00:19:12.406 --> 00:19:18.802 36 will divide by 9 and 36 will divide by 4, so lets multiply 00:19:18.802 --> 00:19:25.731 throughout by that number 36. We have 36, because we've put the 00:19:25.731 --> 00:19:30.528 brackets in we are quite clear that we're multiplying 00:19:30.528 --> 00:19:37.457 everything over that 6 by the 36. Minus 36 times x plus one. 00:19:37.480 --> 00:19:44.350 All over 9 equals 36 times by x, plus three all over 4, so we 00:19:44.350 --> 00:19:49.846 made it quite clear by using the brackets what this 36 is 00:19:49.846 --> 00:19:57.286 multiplying . 6 into six goes once and six into 36 goes 6 times. 00:19:57.286 --> 00:20:01.230 9 into nine goes once and 9 into 36 goes 4. 00:20:01.230 --> 00:20:08.427 4 into 4 goes once and four into 36 goes 9. 00:20:09.160 --> 00:20:14.711 So now I have this bracket to multiply by 6 this bracket to 00:20:14.711 --> 00:20:20.262 multiply by 4 and this bracket to multiply by 9. I don't have 00:20:20.262 --> 00:20:25.386 to worry about the ones because I'm dividing by them so they 00:20:25.386 --> 00:20:29.656 leave everything unchanged. So let's multiply out six times by 00:20:29.656 --> 00:20:35.634 6X plus 30 (6 times by 5). This is a minus four I'm 00:20:35.634 --> 00:20:38.623 multiplied by, so I need to be a 00:20:38.623 --> 00:20:45.330 bit careful. Minus four times x is minus 4x, 00:20:45.330 --> 00:20:48.240 Minus 4 times 1 is minus 4 00:20:48.870 --> 00:20:52.102 Equals 9X and 9 threes 00:20:52.102 --> 00:20:59.310 are 27. So now we need to simplify this side 00:20:59.310 --> 00:21:06.390 6x takeaway 4x. That's just two X30 takeaway, four is 26, and 00:21:06.390 --> 00:21:09.930 that's equal to 9X plus 27. 00:21:11.150 --> 00:21:17.492 Let me take 2X away from each side, so I have 26 equals 7X 00:21:17.492 --> 00:21:23.381 plus 27 and now I'll take the Seven away from each side and 00:21:23.381 --> 00:21:30.176 I'll have minus one is equal to 7X and so now I need to divide 00:21:30.176 --> 00:21:36.518 both sides by 7 and so I get minus 7th for my answer. Don't 00:21:36.518 --> 00:21:41.048 worry that this is a fraction, sometimes they workout like 00:21:41.048 --> 00:21:44.129 that. Don't worry, that is the negative number. Sometimes they 00:21:44.129 --> 00:21:47.621 workout like that. Let's have a look at another one 'cause this 00:21:47.621 --> 00:21:51.695 is a process that you're going to have to be able to do quite 00:21:51.695 --> 00:21:58.238 complicated questions. So we'll take 4 - 5 X. 00:21:58.760 --> 00:22:00.008 All over 6. 00:22:00.850 --> 00:22:04.665 Minus 1 - 2 X all over 00:22:04.665 --> 00:22:10.068 3. Equals 13 over 42. 00:22:11.170 --> 00:22:16.450 What are we going to do? First of all, let's remind ourselves 00:22:16.450 --> 00:22:17.770 that this line. 00:22:19.150 --> 00:22:21.038 Not only means divide. 00:22:21.760 --> 00:22:28.800 Divide 4 - 5 X by 6 but it means divide all of 4 - 5 X by 6. So 00:22:28.800 --> 00:22:33.024 let's put it in a bracket to remind ourselves and let's do 00:22:33.024 --> 00:22:34.080 the same there. 00:22:34.810 --> 00:22:40.150 Now we need a common denominator, six and three and 00:22:40.150 --> 00:22:47.626 40. Two, well, six goes into 42 and three goes into 42 as well. 00:22:47.626 --> 00:22:52.432 So let's choose 42 as our denominator, an multiply 00:22:52.432 --> 00:22:59.374 everything by 42. So will have 42 * 4 - 5 X or 00:22:59.374 --> 00:23:05.782 over 6 - 42 * 1 - 2 X all over 3. 00:23:06.350 --> 00:23:11.888 Equals 42 * 13 over 42. 00:23:14.100 --> 00:23:19.440 Six goes into six once and six goes into 42 Seven times. 00:23:20.400 --> 00:23:26.670 Three goes into three once and free goes into 4214 times. 00:23:27.260 --> 00:23:30.908 42 goes into itself once and 00:23:30.908 --> 00:23:35.537 again once. So now I need to multiply out these brackets. 00:23:36.220 --> 00:23:37.860 And simplify this side. 00:23:38.700 --> 00:23:45.252 So 7 times by 4 gives us 28 Seven times. My minus five 00:23:45.252 --> 00:23:52.308 gives us minus 35 X. Now here we have a minus sign and the 00:23:52.308 --> 00:23:59.364 14, so it's minus 14 times by 1 - 14 and then it's minus 00:23:59.364 --> 00:24:05.916 14 times by minus 2X. So it's plus 28X. Remember we've got to 00:24:05.916 --> 00:24:10.452 take extra care when we've got those minus signs. 00:24:10.510 --> 00:24:17.398 One times by 13 is just 13. Now we need to tidy 00:24:17.398 --> 00:24:24.286 this side up 28 takeaway 14 is just 1435 - 35 X 00:24:24.286 --> 00:24:30.600 plus 28X. Or what's the difference there? It is 7 so 00:24:30.600 --> 00:24:37.488 it's minus Seven X equals 30. Take the 14 away from each 00:24:37.488 --> 00:24:40.932 side. We've minus Seven X equals 00:24:40.932 --> 00:24:46.660 minus one. And divide both sides by minus 7 - 1 divided 00:24:46.660 --> 00:24:51.654 by minus Seven is just a 7th again fractional answer, but 00:24:51.654 --> 00:24:53.016 not to worry. 00:24:54.400 --> 00:25:01.135 When we looked at these, now we want to have a look at a type 00:25:01.135 --> 00:25:04.727 of equation which occasionally causes problems. This particular 00:25:04.727 --> 00:25:07.870 equation or kind of equation looks relatively 00:25:07.870 --> 00:25:12.320 straightforward. Translate numbers get rather more 00:25:12.320 --> 00:25:16.506 difficult. It can cause difficulties, so we got three 00:25:16.506 --> 00:25:19.098 over 5 equals 6 over X 00:25:19.098 --> 00:25:23.140 straightforward. But what do we do? Let's think about it. First 00:25:23.140 --> 00:25:27.027 of all, in terms of fractions, this is a fraction 3/5, and it 00:25:27.027 --> 00:25:28.522 is equal to another fraction 00:25:28.522 --> 00:25:34.141 which is 6. Well family obviously 3/5 is the same 00:25:34.141 --> 00:25:40.537 fraction as 6/10 and so therefore X has got to be equal 00:25:40.537 --> 00:25:44.747 to 10. That's not going to happen with every question. It's 00:25:44.747 --> 00:25:46.771 not going to be as easy as that. 00:25:47.420 --> 00:25:51.645 We're going to have to juggle with the numbers, so how do we 00:25:51.645 --> 00:25:55.545 do that? Well, again, we need a common denominator. We need a 00:25:55.545 --> 00:25:59.770 number that will be divisible by 5 and a number that will be 00:25:59.770 --> 00:26:01.720 divisible by X. And the obvious 00:26:01.720 --> 00:26:04.875 choice is 5X. So let us 00:26:04.875 --> 00:26:12.247 multiply. Both sides by 5X. So we 5X times by 00:26:12.247 --> 00:26:19.277 3/5 equals 5X times by 6 over X and now 00:26:19.277 --> 00:26:26.307 five goes into five once and five goes into five 00:26:26.307 --> 00:26:33.027 once there. X goes into X once an X goes into X. Once 00:26:33.027 --> 00:26:37.834 there, let's remember that we're multiplying by these, so we have 00:26:37.834 --> 00:26:43.515 one times X times three. That's 3X and divided by one, so it's 00:26:43.515 --> 00:26:50.507 still three X equals 5 1 6, which is just 30 and divided by 00:26:50.507 --> 00:26:52.692 one, so it's still 30. 00:26:53.270 --> 00:26:59.270 3X is equal to 30, so X must be equal to 10, which is what we 00:26:59.270 --> 00:27:03.983 had before. Is there another way of looking at this equation? 00:27:03.983 --> 00:27:05.331 Well, yes there is. 00:27:05.840 --> 00:27:13.232 If two fractions are equal that way up there also equal the 00:27:13.232 --> 00:27:15.080 other way up. 00:27:16.890 --> 00:27:21.390 This makes it easier still because all that we need to do 00:27:21.390 --> 00:27:25.890 now is multiply by the common denominator and we can see what 00:27:25.890 --> 00:27:29.640 that common denominator is. It's quite clearly 6 because six 00:27:29.640 --> 00:27:32.640 divides into six and three divides into 6. 00:27:32.650 --> 00:27:39.748 So we had five over three is equal to X over 6, and we're 00:27:39.748 --> 00:27:45.832 going to multiply by this common denominator of six. So six goes 00:27:45.832 --> 00:27:51.916 into six once on each occasion, three goes into three once and 00:27:51.916 --> 00:27:59.014 into six twice, so again X is equal to 10 two times by 5, 00:27:59.014 --> 00:28:01.042 one times by X. 00:28:01.610 --> 00:28:03.510 Whichever way you use. 00:28:04.540 --> 00:28:08.768 Doesn't matter. They should come out the same. There's no reason 00:28:08.768 --> 00:28:12.272 why they shouldn't, but you do have to be careful. The number 00:28:12.272 --> 00:28:15.776 work can be a bit tricky sometimes. Let's have a look at 00:28:15.776 --> 00:28:16.944 just a couple more. 00:28:17.480 --> 00:28:23.666 Five over 3X is equal to 00:28:23.666 --> 00:28:26.759 25 over 27. 00:28:28.060 --> 00:28:34.904 OK. What I think I'm going to do with these is flip them over 00:28:34.904 --> 00:28:41.036 3X over 5 is equal to 27 over 25. I can see straight away. 00:28:41.036 --> 00:28:46.292 I've got a common denominator here of 25. Five goes into 25 00:28:46.292 --> 00:28:52.862 exactly and so does 25. So if I do that multiplication 25 * 3 X 00:28:52.862 --> 00:28:56.804 over 5 is equal to 25 * 27 over 00:28:56.804 --> 00:29:03.424 25. 25 goes into itself once on each occasion. 00:29:04.750 --> 00:29:10.145 Five goes into itself once and five goes into 25 five times. So 00:29:10.145 --> 00:29:17.200 I have 15X5 times by three. X is equal to 27, and so X is 27 over 00:29:17.200 --> 00:29:22.180 15. Dividing both sides by 15 and there is here a common 00:29:22.180 --> 00:29:27.575 factor between top and bottom of three, which gives me 9 over 5, 00:29:27.575 --> 00:29:31.725 so that's an acceptable answer because it's in its lowest 00:29:31.725 --> 00:29:34.630 forms. Or I could write it as 00:29:34.630 --> 00:29:38.932 one. And four fifths, which is also an acceptable answer. 00:29:40.060 --> 00:29:45.716 Now, some of you may not like what I did there when I flipped 00:29:45.716 --> 00:29:51.372 it over, and we might want to think, well, how would I do it 00:29:51.372 --> 00:29:56.624 if I had to start from there. So let's tackle that in another 00:29:56.624 --> 00:30:02.684 way. So again we five over 3X is equal to 25 over 27 this time. 00:30:03.220 --> 00:30:08.164 We're not going to flip it over. Let's look for a common 00:30:08.164 --> 00:30:12.284 denominator here between these two, so we want something that 00:30:12.284 --> 00:30:16.816 3X will divide into exactly, and something that 27 will divide 00:30:16.816 --> 00:30:22.172 into exactly, well. Three will divide into 27, so the 27, so to 00:30:22.172 --> 00:30:28.352 speak ought to be a part of our answer. What we need is an X, 00:30:28.352 --> 00:30:34.532 because if we had 27 X, 3X would divide into it 9 times and the 00:30:34.532 --> 00:30:37.004 27 would just divide into it. 00:30:37.090 --> 00:30:43.844 X times, so that's going to be our common denominator. The 00:30:43.844 --> 00:30:51.212 thing that we are going to multiply both sides by 27 X. 00:30:51.400 --> 00:30:58.420 So we can look at this and we can see that X divides into X 00:30:58.420 --> 00:31:04.972 one St X device into X. Once there we can also see that three 00:31:04.972 --> 00:31:10.120 divides into three and three divides into 27 nine times and 00:31:10.120 --> 00:31:17.140 over here 27 goes into 27 once each time. So I have 9 * 1 00:31:17.140 --> 00:31:19.948 * 5 and 95 S 45. 00:31:20.750 --> 00:31:27.365 Divided by 1 * 1, which is one, so it's still 45 equals 1 times 00:31:27.365 --> 00:31:35.077 by 25. And times by the X there 25 X. Now I need to divide 00:31:35.077 --> 00:31:40.825 both sides by 25 so I have 45 over 25 equals X. 00:31:41.530 --> 00:31:47.014 This is not in its lowest terms. I can divide top on 00:31:47.014 --> 00:31:52.955 bottom by 5, giving me 9 over 5 again, which again I can 00:31:52.955 --> 00:31:55.697 write as one and four fifths. 00:31:57.840 --> 00:32:00.840 Let's take one final example. 00:32:01.440 --> 00:32:06.130 This time, let's look at some fractions, but this time, 00:32:06.130 --> 00:32:11.289 mysteriously, the X is already on the top. That's really good 00:32:11.289 --> 00:32:17.386 for us. All we need to do is look at what's our common 00:32:17.386 --> 00:32:18.793 denominator, well, 7. 00:32:19.960 --> 00:32:26.620 And 49 what number divides exactly by both of these and it 00:32:26.620 --> 00:32:34.390 will be 49. So we just need to do 49 times by 19 X 00:32:34.390 --> 00:32:41.605 over 7 equals 49 * 57 over 4949 goes into 49 once each 00:32:41.605 --> 00:32:46.776 time. And Seven goes into 49 Seven times. 00:32:47.380 --> 00:32:53.230 And so I have 7 times by 19 X equals 57 and you might say, 00:32:53.230 --> 00:32:56.740 well, hang on a minute, shouldn't you have multiplied 00:32:56.740 --> 00:33:01.810 out that first? Well, I didn't want to. Why didn't I want to? 00:33:01.810 --> 00:33:05.320 Well, sometimes when you play darts, your arithmetic improves 00:33:05.320 --> 00:33:10.780 and triple 19 on a dartboard is 57. So 19 divides into 57 three 00:33:10.780 --> 00:33:15.460 times, and I don't want to lose that relationship. So I'm going 00:33:15.460 --> 00:33:18.580 to divide each side by 19 so 7X. 00:33:18.650 --> 00:33:24.302 Is equal to three, which means X must be 3 over 7. 00:33:24.980 --> 00:33:28.472 Playing darts does help with arithmetic. We finished there 00:33:28.472 --> 00:33:31.964 with simple linear equations. The important thing in dealing 00:33:31.964 --> 00:33:36.620 with these kinds of equations and any kind of equation is to 00:33:36.620 --> 00:33:41.664 remember that the equal sign is a balance. What it tells you is 00:33:41.664 --> 00:33:46.320 that what's on the left hand side is exactly equal to what's 00:33:46.320 --> 00:33:51.364 on the right hand side. So whatever you do to one side, you 00:33:51.364 --> 00:33:55.632 have to do to the other side, and you must follow. 00:33:55.650 --> 00:33:58.674 The rules of arithmetic when you do it.