1 00:00:00,820 --> 00:00:05,488 In this session we're going to be having a look at simple equations 2 00:00:05,488 --> 00:00:09,378 in one variable and the equations will be linear. 3 00:00:09,378 --> 00:00:11,712 That means that there are no x-squared terms 4 00:00:11,712 --> 00:00:18,371 and no x-cube terms, just x's and numbers. So let's have a 5 00:00:18,371 --> 00:00:25,286 look at the first one. 3x plus 15 equals x plus 25. 6 00:00:25,286 --> 00:00:29,896 An important thing to remember about any equation is this equal 7 00:00:29,896 --> 00:00:34,967 sign represents a balance. What that equal sign says is that 8 00:00:34,967 --> 00:00:40,960 what's on the left hand side is exactly the same as what's on 9 00:00:40,960 --> 00:00:43,265 the right hand side. 10 00:00:43,310 --> 00:00:48,680 If we do anything to one side of the equation, we have to do it 11 00:00:48,680 --> 00:00:52,618 to the other side. If we don't, the balance is disturbed. 12 00:00:53,370 --> 00:00:55,106 If we can keep that in mind, 13 00:00:55,870 --> 00:00:59,740 the whatever operations we perform on either side of the 14 00:00:59,740 --> 00:01:04,771 equation, so long as it's done in exactly the same way on other side, 15 00:01:04,771 --> 00:01:06,706 we should be alright. 16 00:01:07,520 --> 00:01:12,248 Our first step in solving any equation is to attempt to gather 17 00:01:12,248 --> 00:01:16,188 all the x terms together and all these free floating numbers together. 18 00:01:16,188 --> 00:01:22,098 To begin with we've got 3x on the left and and x on the right. 19 00:01:22,098 --> 00:01:28,402 If we take an x away from both sides, we take one x off the left, 20 00:01:28,402 --> 00:01:34,312 and one x off the right. That will give us 2x plus 15 equals 25. 21 00:01:34,312 --> 00:01:38,252 We need to get the numbers together. These free numbers. 22 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 23 00:01:42,960 --> 00:01:47,580 then we will have no numbers remaining on the left, just a 2x 24 00:01:47,580 --> 00:01:51,870 and taking 15 away on the right, that gives us 10. 25 00:01:51,870 --> 00:01:53,850 So, x must be equal to 5. 26 00:01:54,630 --> 00:01:57,479 That was relatively straightforward. 27 00:01:57,479 --> 00:02:01,956 Lots of plus signs, no minus signs, which we know can be complicated, 28 00:02:01,956 --> 00:02:05,619 also no brackets. So let's introduce both of those. 29 00:02:06,650 --> 00:02:14,190 We will take 2x plus 3 is equal to 30 00:02:14,190 --> 00:02:17,206 6 minus bracket 2x minus three bracket. 31 00:02:18,410 --> 00:02:23,630 Now this right hand side needs a little bit of dealing with. 32 00:02:23,630 --> 00:02:28,850 It needs getting into shape so we have to remove 33 00:02:28,850 --> 00:02:34,940 this bracket. 2x plus 3 equals 6, 34 00:02:34,940 --> 00:02:39,725 now we take away 2x, thats minus 2x and then we're taking away minus 3. 35 00:02:39,725 --> 00:02:44,510 When we take away minus 3, that makes it plus 3. 36 00:02:45,080 --> 00:02:49,724 Before we go any further, we need to tidy this right hand 37 00:02:49,724 --> 00:02:55,916 side up a little bit more 2x plus 3 is equal to 9 minus 2x 38 00:02:55,916 --> 00:03:00,947 and now we're in the same position at this line as we were 39 00:03:00,947 --> 00:03:05,591 when we started there. So we need to get the xs together, 40 00:03:05,591 --> 00:03:11,396 which we can best do by adding 2x to each side. 41 00:03:11,396 --> 00:03:15,266 On the right, minus 2x plus 2x. This side gives us no x. 42 00:03:15,290 --> 00:03:22,235 And on the left it's going to give us 4X. 43 00:03:22,235 --> 00:03:29,180 4X plus 3 is equal to 9 because minus 2X plus 2X gives no x. 44 00:03:29,180 --> 00:03:35,199 Now I can take 3 away from each side, 4X equals 6 45 00:03:35,199 --> 00:03:42,144 and so x is 6 divided by 4 and we want to write that in its 46 00:03:42,144 --> 00:03:46,774 lowest terms which is 3 over 2. Perfectly acceptable answer. 47 00:03:46,780 --> 00:03:51,448 Or one and a half. So either of these are acceptable answers. 48 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 49 00:03:56,116 --> 00:04:01,173 to its lowest terms. So either of those two are OK as answers. 50 00:04:01,750 --> 00:04:06,040 With all of these equations, we should strictly check back by 51 00:04:06,040 --> 00:04:10,720 taking our answer and putting it into the first line of the 52 00:04:10,720 --> 00:04:15,790 equation and seeing if we get the right answer. 53 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 54 00:04:22,420 --> 00:04:26,710 The balance that we talked about at the beginning is maintained. 55 00:04:26,710 --> 00:04:31,780 If you take this number, 3 over 2, or one and a half. 56 00:04:31,800 --> 00:04:35,408 And substitute it back into here we should find again that 57 00:04:35,408 --> 00:04:39,344 it gives us the right answer, that both sides give the same value. 58 00:04:39,344 --> 00:04:42,952 They balance. Solet's just do that. 59 00:04:42,952 --> 00:04:47,544 Let us take one and a half. So 2 times one and a half is 3, 60 00:04:48,130 --> 00:04:53,520 plus three is 6, so we've got six on the left side. 61 00:04:53,520 --> 00:04:57,370 Here we have 6, takeaway, now let's deal with this bracket, 62 00:04:57,370 --> 00:05:01,605 just this bracket on its own, nothing else. 2x takeaway, three, that's nothing, 63 00:05:01,605 --> 00:05:06,225 so we are just left with 6. We calculated this side to be 6. 64 00:05:06,225 --> 00:05:11,230 Six equals six, so again, we've got that 65 00:05:11,230 --> 00:05:15,080 balance, so we know that we've got the right answer. 66 00:05:16,250 --> 00:05:21,437 Let's carry on and have a look at one or two questions with 67 00:05:21,437 --> 00:05:25,028 more brackets and this time numbers outside those 68 00:05:25,028 --> 00:05:30,215 brackets as well. So in a sense, what we're doing is we are 69 00:05:30,215 --> 00:05:33,806 increasing the complexity of the equation, but the simple 70 00:05:33,806 --> 00:05:38,993 principles that we've got so far are going to help us out because 71 00:05:38,993 --> 00:05:42,983 no matter how complicated it gets and this does look 72 00:05:42,983 --> 00:05:46,574 complicated, the same ideas work all the time. 73 00:05:46,600 --> 00:05:52,320 We begin by multiplying out the brackets and taking care, 74 00:05:52,320 --> 00:05:58,560 in particular, with any minus signs that come up so 75 00:05:58,560 --> 00:06:05,320 8 times by x is 8x and eight times by minus three is minus 24. 76 00:06:05,320 --> 00:06:09,480 We've multiplied everything inside that bracket by what's 77 00:06:09,480 --> 00:06:16,240 outside, so we've 8 times by x and we've 8 times by minus 3 78 00:06:16,240 --> 00:06:20,730 Now we remove this bracket were taking away 6, 79 00:06:20,730 --> 00:06:24,276 that's minus six and we're taking away minus 2x, 80 00:06:24,920 --> 00:06:30,822 a minus minus gives us a plus. So that's plus two X 81 00:06:30,822 --> 00:06:38,530 equals, two times x here, is 2x, 2 times by 2 is 4. 82 00:06:38,530 --> 00:06:44,950 and now we have to multiply by minus five, so we've got 83 00:06:44,950 --> 00:06:52,440 minus five times by 5. That's minus 25. And minus five times by minus x. 84 00:06:52,440 --> 00:06:54,580 So that's plus 5x. 85 00:06:55,300 --> 00:07:01,007 Each side needs tidying up. We need to look at this side and 86 00:07:01,007 --> 00:07:06,714 gather x terms and numbers together. so with 8x plus 2x that's 10x. 87 00:07:07,370 --> 00:07:13,717 Take away 24, takeaway six, that's taking away 30 altogether. 88 00:07:13,717 --> 00:07:16,602 2x and 5x gives us 7x. 89 00:07:17,370 --> 00:07:22,694 Add on 4 takeaway 25. Now that's the equivalent of taking 90 00:07:22,694 --> 00:07:29,545 away 21. Let's get the xs together. We can take 7x away from each side 91 00:07:29,545 --> 00:07:35,330 so we would have 3x minus 30 there equals just 92 00:07:35,330 --> 00:07:38,000 minus 21 because we've taken that 7x away. 93 00:07:38,000 --> 00:07:44,314 Add the 30 to each side, because minus 30 + 30 gives us 94 00:07:44,314 --> 00:07:50,816 nothing. And add it on over here. So we get three x equals, 95 00:07:50,816 --> 00:07:57,298 30 added to minus 21, just the same as 30 takeaway 21, 96 00:07:57,298 --> 00:07:59,150 the answer is 9. 97 00:07:59,170 --> 00:08:03,295 And three times by something to give us 9 means the 98 00:08:03,295 --> 00:08:08,170 something, the x, must be 3. Again we ought to be able 99 00:08:08,170 --> 00:08:12,670 to take that 3, put it back into this line and see that 100 00:08:12,670 --> 00:08:17,545 in fact we've got the correct answer. So let's just try. 101 00:08:17,545 --> 00:08:22,420 3 minus 3 is zero. 8 times by 0 is still 0, 102 00:08:22,420 --> 00:08:23,545 so we can forget about that term. 103 00:08:25,090 --> 00:08:29,640 2 times by 3three is 6, so six takeaway six is nothing, so 104 00:08:29,640 --> 00:08:33,140 we've actually got nothing there, zero, and nothing there, zero. 105 00:08:33,140 --> 00:08:37,340 so there's nothing on the left side of the equation. 106 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. 107 00:08:42,590 --> 00:08:48,190 3 plus 2 is 5 and 2 fives are 10. 5 takeaway 3 is 2, 108 00:08:48,190 --> 00:08:53,090 and 5 twos are 10. So we've got 10 takeaway 10, 109 00:08:53,090 --> 00:08:55,190 this gives us nothing again, so we've got nothing on the right. 110 00:08:55,190 --> 00:09:00,334 The equation balances with this value, 111 00:09:00,334 --> 00:09:03,082 so this is our one and only solution. 112 00:09:03,082 --> 00:09:10,300 We take a final example of this kind with 113 00:09:10,300 --> 00:09:15,820 x plus 1 times by 2x plus 1. 114 00:09:16,370 --> 00:09:23,954 Equals x plus 3 times by 2x plus 3 115 00:09:23,954 --> 00:09:26,798 - 14. 116 00:09:27,590 --> 00:09:31,869 Now, I did say that these were linear equations that there 117 00:09:31,869 --> 00:09:34,592 would be no x-squared terms in them. 118 00:09:34,592 --> 00:09:38,322 But when we start to multiply out these 119 00:09:38,322 --> 00:09:41,612 brackets, we are going to get some x-squared terms. 120 00:09:42,990 --> 00:09:49,150 Let's have a look, to show you what actually does happen. 121 00:09:49,150 --> 00:09:52,230 We do x times by two x. That's two x-squared. 122 00:09:52,230 --> 00:09:57,880 x times by one, that is plus x. 123 00:09:58,530 --> 00:10:01,785 One times two x, that is plus 2x. 124 00:10:01,785 --> 00:10:05,715 1 times by 1 125 00:10:05,715 --> 00:10:08,810 + 1. Equals... 126 00:10:10,110 --> 00:10:13,926 x times by two x, that's two x-squared. 127 00:10:15,790 --> 00:10:19,500 x times by three is 3x.. 128 00:10:20,190 --> 00:10:23,640 3 times by 2x is 6x. 129 00:10:24,490 --> 00:10:30,001 3 times by 3 is 9 and finally takeaway 14. 130 00:10:30,001 --> 00:10:35,011 Now we need to tidy up both sides here. 131 00:10:35,870 --> 00:10:42,710 Two x-squared plus 3x plus one equals 132 00:10:42,710 --> 00:10:48,182 2 x-squared plus 9x, (taking these two terms together) 133 00:10:48,820 --> 00:10:52,448 Plus 9, minus 14. 134 00:10:54,510 --> 00:10:58,709 And this needs a little bit more tidying, but before we do that, 135 00:10:58,709 --> 00:11:03,231 let's just have a look at what we've got. 136 00:11:03,231 --> 00:11:06,784 We've got 2 x-squared there, and 2 x-squared there. They are both 137 00:11:06,784 --> 00:11:10,660 positive, so I can take two x-sqaured away from both sides. 138 00:11:10,660 --> 00:11:14,213 That means that the two x-squared vanishes from both 139 00:11:14,213 --> 00:11:18,089 sides, so let's do that. Take two x-squared from each side, 140 00:11:18,089 --> 00:11:22,934 and that leaves us 3x plus one, is equal to 9x, and now we can 141 00:11:22,934 --> 00:11:24,872 do this bit 9 takeaway 14, 142 00:11:24,940 --> 00:11:29,417 Well, that's going to be takeaway five. I still have the 143 00:11:29,417 --> 00:11:34,708 five to subtract. Now we're back to where we used to be getting 144 00:11:34,708 --> 00:11:38,371 the xs together, getting the numbers together. We can 145 00:11:38,371 --> 00:11:43,662 subtract 3x from both sides, so that gives me one equals 6x minus 5 146 00:11:43,662 --> 00:11:49,360 We can add the five to both sides so that 147 00:11:49,360 --> 00:11:52,616 6 equals 6X, and so 1 is equal to x. 148 00:11:53,810 --> 00:11:58,118 And again, we can check that this works. We can substitute it back in. 149 00:11:58,118 --> 00:12:03,862 1 plus 1 is 2, two ones are two, and one is 3. 150 00:12:03,862 --> 00:12:08,529 So effectively we've got 3 times by two at this side, which gives 6. 151 00:12:08,529 --> 00:12:16,045 Here 1 + 3 is 4 two times by one is 2, + 3 is 5 152 00:12:16,045 --> 00:12:21,388 so we've got four times by 5. So altogether there there's 20. 153 00:12:22,100 --> 00:12:26,940 Takeaway 14 is again 6, so again, this equation is balanced 154 00:12:26,940 --> 00:12:31,340 exactly when we take x to be equal to 1. 155 00:12:32,410 --> 00:12:36,700 Now, those equations that we've just looked at have really been 156 00:12:36,700 --> 00:12:39,820 about whole numbers. The coefficients have been whole 157 00:12:39,820 --> 00:12:43,330 numbers. Everything's been in terms of integers. What happens 158 00:12:43,330 --> 00:12:47,230 when we start to get some fractions in there? Some 159 00:12:47,230 --> 00:12:51,910 rational numbers? How do we deal with that? So again, we're going 160 00:12:51,910 --> 00:12:56,590 to add in more complexity, but again, the rules are the same. 161 00:12:56,590 --> 00:13:02,440 What we do to one side of the equation we must do to the other 162 00:13:02,440 --> 00:13:04,780 side in order to preserve that 163 00:13:04,780 --> 00:13:10,930 particular balance. So let's introduce some fractions 164 00:13:10,930 --> 00:13:14,068 along with some brackets. 165 00:13:14,068 --> 00:13:22,050 4 bracket X plus 2, all over 5 is equal to 166 00:13:22,050 --> 00:13:25,470 7 plus 5x over 13. 167 00:13:26,180 --> 00:13:28,950 We've got some fractions here. 168 00:13:29,790 --> 00:13:35,404 Numbers in the denominator 5 and 30. We want rid of those we want 169 00:13:35,404 --> 00:13:40,216 to be able to work with whole numbers with integers, so we 170 00:13:40,216 --> 00:13:45,830 have to find a way of getting rid of them. That means we 171 00:13:45,830 --> 00:13:50,241 have to multiply everything because what we do to one side, 172 00:13:50,241 --> 00:13:55,053 we must do to the other. 173 00:13:55,053 --> 00:13:59,865 The common denominator for five and 13 is 65, that's five times by 13. 174 00:13:59,865 --> 00:14:05,080 So let's do that. Let's multiply everything by 65 and 175 00:14:05,080 --> 00:14:10,778 I'll write it down in full so that we can see it happening. 176 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. 177 00:14:17,290 --> 00:14:22,174 Remember, I said we have to multiply everything, so it's not 178 00:14:22,174 --> 00:14:26,651 just these fraction bits, it's any spare numbers that there are 179 00:14:26,651 --> 00:14:27,872 around as well. 180 00:14:28,390 --> 00:14:34,550 Plus 65 times by 5x over 30. 181 00:14:35,260 --> 00:14:41,070 Now let's look at each term and make it simpler. Tidy it up. 182 00:14:41,070 --> 00:14:46,880 For a start, five will divide into 65, so 5 into five goes one 183 00:14:46,880 --> 00:14:49,370 and five into 65 goes 13 times. 184 00:14:50,110 --> 00:14:57,950 And then four times by 13? Well, that's 52, so we have 185 00:14:57,950 --> 00:15:01,870 52 times by x +2 equals. 186 00:15:02,790 --> 00:15:07,422 That's in a nice familiar form where used to that sort of 187 00:15:07,422 --> 00:15:11,282 former. We've arrived at it by choosing to multiply everything 188 00:15:11,282 --> 00:15:14,756 by this common denominator. Let's tidy this side up. 189 00:15:15,490 --> 00:15:21,250 7 times by 65. Well, that's pretty tall order. Let's try it. 190 00:15:21,250 --> 00:15:26,050 Seven 5s are 35. Seven 6s are 42 and three is 45. 191 00:15:26,760 --> 00:15:33,956 Here with 65 times 5 x over 13. 13 goes into 13 once, 192 00:15:33,956 --> 00:15:41,152 and 13 goes into 65 five times. So we five times by 5x is 25x. 193 00:15:41,152 --> 00:15:46,334 You may say 'what happened to these ones?'. Well if I divide by 194 00:15:46,334 --> 00:15:49,958 one it stays unchanged so I don't have to write them down. 195 00:15:50,970 --> 00:15:56,885 And now we left with an equation that were used to 196 00:15:56,885 --> 00:16:01,435 handling. We've met these kind before, so let's multiply out 197 00:16:01,435 --> 00:16:05,985 the brackets, get the xs together and solve the equation. 198 00:16:05,985 --> 00:16:12,355 So we multiply out the brackets 52 times 2 is 104 199 00:16:12,355 --> 00:16:19,180 equals 455 plus 25x. Take the 25x away from each side, 200 00:16:19,180 --> 00:16:21,455 gives me 27x there. 201 00:16:21,470 --> 00:16:27,509 And no x is there. Take the 104 away 202 00:16:27,509 --> 00:16:32,206 from each side, which gives me 351. 203 00:16:33,410 --> 00:16:38,701 And now lots of big numbers, really, that shouldn't be a 204 00:16:38,701 --> 00:16:43,511 problem to us. 351 divided by 27 will certainly go once 205 00:16:44,190 --> 00:16:48,782 (there's one 27 in the 35 and eight over and 27s into 8 206 00:16:48,782 --> 00:16:53,374 go three times, so answer is x equals 13 and we 207 00:16:53,374 --> 00:16:57,310 should go back and check it and make sure that it's right. 208 00:16:57,310 --> 00:16:58,622 So let's do that. 209 00:16:59,460 --> 00:17:02,815 13 + 2? Well, that's 15 210 00:17:02,815 --> 00:17:09,818 15 divided 5 is 3 and now times by 4, so that's 12. 211 00:17:09,818 --> 00:17:15,306 Hang on to that number 12 at that side. 212 00:17:15,306 --> 00:17:20,402 5 times 13 divided by 13. So the answer is just five 213 00:17:20,402 --> 00:17:25,890 plus the 7 is again 12. The same as this side. 214 00:17:25,890 --> 00:17:29,418 So the answer is correct. It balances. 215 00:17:30,190 --> 00:17:35,151 Let's practice that one again and have a look at another example. 216 00:17:35,151 --> 00:17:39,086 We take X plus 5 over 6 217 00:17:39,086 --> 00:17:45,938 minus x plus one over 9 218 00:17:46,440 --> 00:17:50,262 is equal to x plus three over 4. 219 00:17:50,262 --> 00:17:53,842 We haven't got any brackets. 220 00:17:53,842 --> 00:17:58,077 Does that make any difference? The thing you have 221 00:17:58,077 --> 00:18:00,075 to remember is that this line. 222 00:18:00,930 --> 00:18:04,700 Not only acts as a division sign, but it acts as a bracket. 223 00:18:05,230 --> 00:18:11,665 It means that all of x plus 5 is divided by 6. 224 00:18:12,410 --> 00:18:18,080 So it might be as well if we kept that in mind and put 225 00:18:18,080 --> 00:18:22,535 brackets around these terms so that we're clear that 226 00:18:22,535 --> 00:18:27,395 we've written down that these are to be kept together and are 227 00:18:27,395 --> 00:18:32,660 all divided by 6. These two are to be kept together and all 228 00:18:32,660 --> 00:18:37,115 divided by 9. And similarly here they are all divided by 4. 229 00:18:37,970 --> 00:18:42,799 Next step we need a common denominator. We need a number 230 00:18:42,799 --> 00:18:47,628 into which all of these will divide exactly. Now we could 231 00:18:47,628 --> 00:18:50,701 multiply them altogether and we'd be certain. 232 00:18:51,250 --> 00:18:57,581 But the arithmetic would be horrendous. 6 times 9 times 4 is very big. 233 00:18:57,581 --> 00:19:03,425 Can we find a smaller number into which six, nine, and 234 00:19:03,425 --> 00:19:05,373 four will all divide? 235 00:19:06,010 --> 00:19:12,406 Well, a candidate for that is 36. 36 will divide by 6l 236 00:19:12,406 --> 00:19:18,802 36 will divide by 9 and 36 will divide by 4, so lets multiply 237 00:19:18,802 --> 00:19:25,731 throughout by that number 36. We have 36, because we've put the 238 00:19:25,731 --> 00:19:30,528 brackets in we are quite clear that we're multiplying 239 00:19:30,528 --> 00:19:37,457 everything over that 6 by the 36. Minus 36 times x plus one. 240 00:19:37,480 --> 00:19:44,350 All over 9 equals 36 times by x, plus three all over 4, so we 241 00:19:44,350 --> 00:19:49,846 made it quite clear by using the brackets what this 36 is 242 00:19:49,846 --> 00:19:57,286 multiplying . 6 into six goes once and six into 36 goes 6 times. 243 00:19:57,286 --> 00:20:01,230 9 into nine goes once and 9 into 36 goes 4. 244 00:20:01,230 --> 00:20:08,427 4 into 4 goes once and four into 36 goes 9. 245 00:20:09,160 --> 00:20:14,711 So now I have this bracket to multiply by 6 this bracket to 246 00:20:14,711 --> 00:20:20,262 multiply by 4 and this bracket to multiply by 9. I don't have 247 00:20:20,262 --> 00:20:25,386 to worry about the ones because I'm dividing by them so they 248 00:20:25,386 --> 00:20:29,656 leave everything unchanged. So let's multiply out six times by 249 00:20:29,656 --> 00:20:35,634 6X plus 30 (6 times by 5). This is a minus four I'm 250 00:20:35,634 --> 00:20:38,623 multiplied by, so I need to be a 251 00:20:38,623 --> 00:20:45,330 bit careful. Minus four times x is minus 4x, 252 00:20:45,330 --> 00:20:48,240 Minus 4 times 1 is minus 4 253 00:20:48,870 --> 00:20:52,102 Equals 9X and 9 threes 254 00:20:52,102 --> 00:20:59,310 are 27. So now we need to simplify this side 255 00:20:59,310 --> 00:21:06,390 6x takeaway 4x. That's just two X30 takeaway, four is 26, and 256 00:21:06,390 --> 00:21:09,930 that's equal to 9X plus 27. 257 00:21:11,150 --> 00:21:17,492 Let me take 2X away from each side, so I have 26 equals 7X 258 00:21:17,492 --> 00:21:23,381 plus 27 and now I'll take the Seven away from each side and 259 00:21:23,381 --> 00:21:30,176 I'll have minus one is equal to 7X and so now I need to divide 260 00:21:30,176 --> 00:21:36,518 both sides by 7 and so I get minus 7th for my answer. Don't 261 00:21:36,518 --> 00:21:41,048 worry that this is a fraction, sometimes they workout like 262 00:21:41,048 --> 00:21:44,129 that. Don't worry, that is the negative number. Sometimes they 263 00:21:44,129 --> 00:21:47,621 workout like that. Let's have a look at another one 'cause this 264 00:21:47,621 --> 00:21:51,695 is a process that you're going to have to be able to do quite 265 00:21:51,695 --> 00:21:58,238 complicated questions. So we'll take 4 - 5 X. 266 00:21:58,760 --> 00:22:00,008 All over 6. 267 00:22:00,850 --> 00:22:04,665 Minus 1 - 2 X all over 268 00:22:04,665 --> 00:22:10,068 3. Equals 13 over 42. 269 00:22:11,170 --> 00:22:16,450 What are we going to do? First of all, let's remind ourselves 270 00:22:16,450 --> 00:22:17,770 that this line. 271 00:22:19,150 --> 00:22:21,038 Not only means divide. 272 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 273 00:22:28,800 --> 00:22:33,024 let's put it in a bracket to remind ourselves and let's do 274 00:22:33,024 --> 00:22:34,080 the same there. 275 00:22:34,810 --> 00:22:40,150 Now we need a common denominator, six and three and 276 00:22:40,150 --> 00:22:47,626 40. Two, well, six goes into 42 and three goes into 42 as well. 277 00:22:47,626 --> 00:22:52,432 So let's choose 42 as our denominator, an multiply 278 00:22:52,432 --> 00:22:59,374 everything by 42. So will have 42 * 4 - 5 X or 279 00:22:59,374 --> 00:23:05,782 over 6 - 42 * 1 - 2 X all over 3. 280 00:23:06,350 --> 00:23:11,888 Equals 42 * 13 over 42. 281 00:23:14,100 --> 00:23:19,440 Six goes into six once and six goes into 42 Seven times. 282 00:23:20,400 --> 00:23:26,670 Three goes into three once and free goes into 4214 times. 283 00:23:27,260 --> 00:23:30,908 42 goes into itself once and 284 00:23:30,908 --> 00:23:35,537 again once. So now I need to multiply out these brackets. 285 00:23:36,220 --> 00:23:37,860 And simplify this side. 286 00:23:38,700 --> 00:23:45,252 So 7 times by 4 gives us 28 Seven times. My minus five 287 00:23:45,252 --> 00:23:52,308 gives us minus 35 X. Now here we have a minus sign and the 288 00:23:52,308 --> 00:23:59,364 14, so it's minus 14 times by 1 - 14 and then it's minus 289 00:23:59,364 --> 00:24:05,916 14 times by minus 2X. So it's plus 28X. Remember we've got to 290 00:24:05,916 --> 00:24:10,452 take extra care when we've got those minus signs. 291 00:24:10,510 --> 00:24:17,398 One times by 13 is just 13. Now we need to tidy 292 00:24:17,398 --> 00:24:24,286 this side up 28 takeaway 14 is just 1435 - 35 X 293 00:24:24,286 --> 00:24:30,600 plus 28X. Or what's the difference there? It is 7 so 294 00:24:30,600 --> 00:24:37,488 it's minus Seven X equals 30. Take the 14 away from each 295 00:24:37,488 --> 00:24:40,932 side. We've minus Seven X equals 296 00:24:40,932 --> 00:24:46,660 minus one. And divide both sides by minus 7 - 1 divided 297 00:24:46,660 --> 00:24:51,654 by minus Seven is just a 7th again fractional answer, but 298 00:24:51,654 --> 00:24:53,016 not to worry. 299 00:24:54,400 --> 00:25:01,135 When we looked at these, now we want to have a look at a type 300 00:25:01,135 --> 00:25:04,727 of equation which occasionally causes problems. This particular 301 00:25:04,727 --> 00:25:07,870 equation or kind of equation looks relatively 302 00:25:07,870 --> 00:25:12,320 straightforward. Translate numbers get rather more 303 00:25:12,320 --> 00:25:16,506 difficult. It can cause difficulties, so we got three 304 00:25:16,506 --> 00:25:19,098 over 5 equals 6 over X 305 00:25:19,098 --> 00:25:23,140 straightforward. But what do we do? Let's think about it. First 306 00:25:23,140 --> 00:25:27,027 of all, in terms of fractions, this is a fraction 3/5, and it 307 00:25:27,027 --> 00:25:28,522 is equal to another fraction 308 00:25:28,522 --> 00:25:34,141 which is 6. Well family obviously 3/5 is the same 309 00:25:34,141 --> 00:25:40,537 fraction as 6/10 and so therefore X has got to be equal 310 00:25:40,537 --> 00:25:44,747 to 10. That's not going to happen with every question. It's 311 00:25:44,747 --> 00:25:46,771 not going to be as easy as that. 312 00:25:47,420 --> 00:25:51,645 We're going to have to juggle with the numbers, so how do we 313 00:25:51,645 --> 00:25:55,545 do that? Well, again, we need a common denominator. We need a 314 00:25:55,545 --> 00:25:59,770 number that will be divisible by 5 and a number that will be 315 00:25:59,770 --> 00:26:01,720 divisible by X. And the obvious 316 00:26:01,720 --> 00:26:04,875 choice is 5X. So let us 317 00:26:04,875 --> 00:26:12,247 multiply. Both sides by 5X. So we 5X times by 318 00:26:12,247 --> 00:26:19,277 3/5 equals 5X times by 6 over X and now 319 00:26:19,277 --> 00:26:26,307 five goes into five once and five goes into five 320 00:26:26,307 --> 00:26:33,027 once there. X goes into X once an X goes into X. Once 321 00:26:33,027 --> 00:26:37,834 there, let's remember that we're multiplying by these, so we have 322 00:26:37,834 --> 00:26:43,515 one times X times three. That's 3X and divided by one, so it's 323 00:26:43,515 --> 00:26:50,507 still three X equals 5 1 6, which is just 30 and divided by 324 00:26:50,507 --> 00:26:52,692 one, so it's still 30. 325 00:26:53,270 --> 00:26:59,270 3X is equal to 30, so X must be equal to 10, which is what we 326 00:26:59,270 --> 00:27:03,983 had before. Is there another way of looking at this equation? 327 00:27:03,983 --> 00:27:05,331 Well, yes there is. 328 00:27:05,840 --> 00:27:13,232 If two fractions are equal that way up there also equal the 329 00:27:13,232 --> 00:27:15,080 other way up. 330 00:27:16,890 --> 00:27:21,390 This makes it easier still because all that we need to do 331 00:27:21,390 --> 00:27:25,890 now is multiply by the common denominator and we can see what 332 00:27:25,890 --> 00:27:29,640 that common denominator is. It's quite clearly 6 because six 333 00:27:29,640 --> 00:27:32,640 divides into six and three divides into 6. 334 00:27:32,650 --> 00:27:39,748 So we had five over three is equal to X over 6, and we're 335 00:27:39,748 --> 00:27:45,832 going to multiply by this common denominator of six. So six goes 336 00:27:45,832 --> 00:27:51,916 into six once on each occasion, three goes into three once and 337 00:27:51,916 --> 00:27:59,014 into six twice, so again X is equal to 10 two times by 5, 338 00:27:59,014 --> 00:28:01,042 one times by X. 339 00:28:01,610 --> 00:28:03,510 Whichever way you use. 340 00:28:04,540 --> 00:28:08,768 Doesn't matter. They should come out the same. There's no reason 341 00:28:08,768 --> 00:28:12,272 why they shouldn't, but you do have to be careful. The number 342 00:28:12,272 --> 00:28:15,776 work can be a bit tricky sometimes. Let's have a look at 343 00:28:15,776 --> 00:28:16,944 just a couple more. 344 00:28:17,480 --> 00:28:23,666 Five over 3X is equal to 345 00:28:23,666 --> 00:28:26,759 25 over 27. 346 00:28:28,060 --> 00:28:34,904 OK. What I think I'm going to do with these is flip them over 347 00:28:34,904 --> 00:28:41,036 3X over 5 is equal to 27 over 25. I can see straight away. 348 00:28:41,036 --> 00:28:46,292 I've got a common denominator here of 25. Five goes into 25 349 00:28:46,292 --> 00:28:52,862 exactly and so does 25. So if I do that multiplication 25 * 3 X 350 00:28:52,862 --> 00:28:56,804 over 5 is equal to 25 * 27 over 351 00:28:56,804 --> 00:29:03,424 25. 25 goes into itself once on each occasion. 352 00:29:04,750 --> 00:29:10,145 Five goes into itself once and five goes into 25 five times. So 353 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 354 00:29:17,200 --> 00:29:22,180 15. Dividing both sides by 15 and there is here a common 355 00:29:22,180 --> 00:29:27,575 factor between top and bottom of three, which gives me 9 over 5, 356 00:29:27,575 --> 00:29:31,725 so that's an acceptable answer because it's in its lowest 357 00:29:31,725 --> 00:29:34,630 forms. Or I could write it as 358 00:29:34,630 --> 00:29:38,932 one. And four fifths, which is also an acceptable answer. 359 00:29:40,060 --> 00:29:45,716 Now, some of you may not like what I did there when I flipped 360 00:29:45,716 --> 00:29:51,372 it over, and we might want to think, well, how would I do it 361 00:29:51,372 --> 00:29:56,624 if I had to start from there. So let's tackle that in another 362 00:29:56,624 --> 00:30:02,684 way. So again we five over 3X is equal to 25 over 27 this time. 363 00:30:03,220 --> 00:30:08,164 We're not going to flip it over. Let's look for a common 364 00:30:08,164 --> 00:30:12,284 denominator here between these two, so we want something that 365 00:30:12,284 --> 00:30:16,816 3X will divide into exactly, and something that 27 will divide 366 00:30:16,816 --> 00:30:22,172 into exactly, well. Three will divide into 27, so the 27, so to 367 00:30:22,172 --> 00:30:28,352 speak ought to be a part of our answer. What we need is an X, 368 00:30:28,352 --> 00:30:34,532 because if we had 27 X, 3X would divide into it 9 times and the 369 00:30:34,532 --> 00:30:37,004 27 would just divide into it. 370 00:30:37,090 --> 00:30:43,844 X times, so that's going to be our common denominator. The 371 00:30:43,844 --> 00:30:51,212 thing that we are going to multiply both sides by 27 X. 372 00:30:51,400 --> 00:30:58,420 So we can look at this and we can see that X divides into X 373 00:30:58,420 --> 00:31:04,972 one St X device into X. Once there we can also see that three 374 00:31:04,972 --> 00:31:10,120 divides into three and three divides into 27 nine times and 375 00:31:10,120 --> 00:31:17,140 over here 27 goes into 27 once each time. So I have 9 * 1 376 00:31:17,140 --> 00:31:19,948 * 5 and 95 S 45. 377 00:31:20,750 --> 00:31:27,365 Divided by 1 * 1, which is one, so it's still 45 equals 1 times 378 00:31:27,365 --> 00:31:35,077 by 25. And times by the X there 25 X. Now I need to divide 379 00:31:35,077 --> 00:31:40,825 both sides by 25 so I have 45 over 25 equals X. 380 00:31:41,530 --> 00:31:47,014 This is not in its lowest terms. I can divide top on 381 00:31:47,014 --> 00:31:52,955 bottom by 5, giving me 9 over 5 again, which again I can 382 00:31:52,955 --> 00:31:55,697 write as one and four fifths. 383 00:31:57,840 --> 00:32:00,840 Let's take one final example. 384 00:32:01,440 --> 00:32:06,130 This time, let's look at some fractions, but this time, 385 00:32:06,130 --> 00:32:11,289 mysteriously, the X is already on the top. That's really good 386 00:32:11,289 --> 00:32:17,386 for us. All we need to do is look at what's our common 387 00:32:17,386 --> 00:32:18,793 denominator, well, 7. 388 00:32:19,960 --> 00:32:26,620 And 49 what number divides exactly by both of these and it 389 00:32:26,620 --> 00:32:34,390 will be 49. So we just need to do 49 times by 19 X 390 00:32:34,390 --> 00:32:41,605 over 7 equals 49 * 57 over 4949 goes into 49 once each 391 00:32:41,605 --> 00:32:46,776 time. And Seven goes into 49 Seven times. 392 00:32:47,380 --> 00:32:53,230 And so I have 7 times by 19 X equals 57 and you might say, 393 00:32:53,230 --> 00:32:56,740 well, hang on a minute, shouldn't you have multiplied 394 00:32:56,740 --> 00:33:01,810 out that first? Well, I didn't want to. Why didn't I want to? 395 00:33:01,810 --> 00:33:05,320 Well, sometimes when you play darts, your arithmetic improves 396 00:33:05,320 --> 00:33:10,780 and triple 19 on a dartboard is 57. So 19 divides into 57 three 397 00:33:10,780 --> 00:33:15,460 times, and I don't want to lose that relationship. So I'm going 398 00:33:15,460 --> 00:33:18,580 to divide each side by 19 so 7X. 399 00:33:18,650 --> 00:33:24,302 Is equal to three, which means X must be 3 over 7. 400 00:33:24,980 --> 00:33:28,472 Playing darts does help with arithmetic. We finished there 401 00:33:28,472 --> 00:33:31,964 with simple linear equations. The important thing in dealing 402 00:33:31,964 --> 00:33:36,620 with these kinds of equations and any kind of equation is to 403 00:33:36,620 --> 00:33:41,664 remember that the equal sign is a balance. What it tells you is 404 00:33:41,664 --> 00:33:46,320 that what's on the left hand side is exactly equal to what's 405 00:33:46,320 --> 00:33:51,364 on the right hand side. So whatever you do to one side, you 406 00:33:51,364 --> 00:33:55,632 have to do to the other side, and you must follow. 407 00:33:55,650 --> 00:33:58,674 The rules of arithmetic when you do it.