We start with the first stage of the topic “Linear equation”. Let’s start by doing a few tasks. Let’s say I have the equation: 5 / this is big, thick 5 / … 5x = 20 This may seem a little unfamiliar to you at first, but if I paraphrase it, you will realize that it is quite an easy task. It’s all the same thing as saying: 5 by ‘question mark’ is equal to 20. The reason we write this is: we write 5 to x, is because when we write a number to a variable, we accept that we multiply them. So that just means ‘5 by x’. Instead of a question mark, we write x. So 5 over x is equal to 20. Now, you can probably do that in your mind. You can say, “Which number 5 is equal to 20?” Well, that’s 4. But I’ll show you a way to do it consistently, in case 5 is a more complex number. Let me make my pen a little thinner. Okay. I’m rewriting it. If I had 5x = 20, we can do 2 things and they are essentially the same thing. We can simply divide both sides of this equation by 5, in which case on the left these two fives will be shortened. We get x. And then on the right: 20 divided by 5 is 4. And we decided it. There is another way to do it and it is actually the same. We just formulate it a little differently. If we say 5x is equal to 20, instead of dividing by 5, we can multiply 1/5. And if you look at that, you might realize that multiplication by 1/5 is the same such as the division of 5, if you know the difference between division and multiplication of fractions. And then that brings us to the same thing. 1/5 by 5 is 1. So, we are left with x is equal to 4. I intend to focus a little more on that, because when we start having fractions instead of fives, it’s easier to just think about multiplication on the reciprocal. In fact, let’s do one of these now. Let’s say I had -3/4 by x, is equal on 10/13. Now this is a more difficult task. I can’t do it in my mind. We say -3/4 by some number x is equal to 10/13. If someone comes to you and asks you how much is that … I think you’ll react like me you will probably be speechless. But let’s do it algebraically. Okay, we’re doing the same thing. Multiply both sides by the coefficient before x. The coefficient – this complex word – means simply ‘the number multiplied by x’. What is the reciprocal of -3/4? It is -4/3 on … the point is another way of representing multiplication. And you’re probably wondering why in algebra there are all these other ways of writing multiplication, on a par with the standard way of spelling it. The main reason is that just the usual sign of multiplication just confused with the variable x, so have also devised to use a point in case we multiply two constants or we simply write it to a variable to indicate that we are multiplying a variable. So if we multiply the left side by -4/3, we have to do the same thing and on the right. -4/3. On the left side -4/3 and -3/4 will be shortened. You can do it yourself and see that it is so. They are equal to one, so we are left with only x … x is equal to 10 to -4 is -40. 13 to 3, this is equal to 39. So we get x is equal to minus 40 over 39. I like to leave fractions wrong, because it is easier to work with them. But you can also – if you want to write this as a mixed number – will become minus 1 and 1/39. I will keep it that way. Let’s check it to make sure it’s correct. The good thing about algebra is that you can always get the answer and put it back in the original equation, to make sure it’s right. The original equation was -3/4 by x, and here we substitute x back into the equation. Where we see x, we will now put the answer. So we replace with -40/39. Our initial equation says that this is equal to 10/13. Okay. Once again, when I write 3/4 right next to the parentheses like this, this is another way to write multiplication. And so, -3 to -40 is a minus … we can actually do something a little simpler. This 4 becomes 1, and this becomes 10. When we multiply fractions, we can simplify in this way. And it becomes plus 30 (because we have minus minus and 3 by 10) on 4 is now 1, so all that’s left is 39. And 30/39, if we divide the numerator and denominator of 3, we get 10/13, which is the same thing which the equation said we would get. So we know we got the right answer. Let’s do another task. Minus 5/6 x is equal to 7/8. And if you want to try to solve this problem on your own, now is a good time to pause. And I’m starting to do the task now. So, the same thing. What is the reciprocal of -5/6? Well, that’s -6/5. We multiply this. If we do it on the right side – we must also do it on the right side. -6/5. On the left, -6/5 and -5/6 are shortened. We are left with only x. And on the right side we have we can divide both 6 and 8 by 2, so this becomes -3, and this happens 4. 7 over -3 is -21, on 20. I accept that I did not make any mistakes out of carelessness. That must be true. In fact, let’s just check it out really quickly. -5/6 at -21/20. Ok this is equal to … we can do this 1, this 4, let’s do this 2, to do this 7. Negative to negative is positive. So 7. 2 by 4 is 8! And that’s what we said we would get. So we get it right. I think you can safely now to try some linear equations from stage 1. Have fun!