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